# Fractal Cities: Then and Now (Version 2)

This unit provides an approach for creating a math investigation that includes reading-focused inquiry to build student math literacy skills. The unit was created by a teacher cohort in year two of the School Librarians Advancing STEM Learning project, led by the Institute for the Study of Knowledge Management (ISKME) in partnership with Granite State University, New Hampshire, and funded by the Institute for Museum and Library Services (IMLS).

### Part I: Unit Title

Fractal Cities: Now and Then

### Part II: Background on LMS and Science Teacher relationship:

This lesson was created by Library Media Specialist (Pam Harland)  and Math teachers (Rebecca Hanna and Carissa Maskwa). Pam’s strengths are identified as text-based inquiry and she requested to see Becky and Carissa’s Common Core Math Standards and school competencies connected to inquiry research and fractals in Geometry. Becky and Carissa’s strengths are Math content knowledge and they requested to see Pam’s Content Area Research Rubric which includes Common Core Reading and Writing standards.

### Part III: Unit Description:

This unit includes 6 unit days. Our periods are 80 minutes long.

Over the course of the unit, students will explore a variety of texts and grow in their knowledge of fractals, city design, and ability to use informational text to support their inquiry and research.

Day 1: Launch with videos introducing the topic of fractals . Ask students if they have ever seen anything like it.

Day 2: Launch Mathigon on Chrome. Manipulate the fractal designs using the Chromebooks. Ask students to reflect on this activity. Share maps of ancient cities with fractals.

Day 3: Using inquiry-focused reading, students will read the anchor text and view the maps with the big questions in mind: category questions, generalizations, and patterns. Students will also investigate the actual map referenced in the text to garner additional information and understanding.

Day 4: Summative assignment in pairs of students: Link fractals to city planning. Student pairs will create their own city design.

Day 5 & 6: Work days and presentation.

CCSS Math Standards

• Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
• Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
• 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.

CCSS Literacy Standards

• CCSS.ELA-LITERACY.RST.9-10.1
Cite specific textual evidence to support analysis of science and technical texts, attending to the precise details of explanations or descriptions.
• CCSS.ELA-LITERACY.RST.9-10.2
Determine the central ideas or conclusions of a text; trace the text's explanation or depiction of a complex process, phenomenon, or concept; provide an accurate summary of the text.

### Part V: Unit Essential Question

How can fractals be used in design of cities today?

### Part VI: Goals for Using Inquiry

The goal for using inquiry in this unit is to have students develop their own supporting research questions around fractals, examine provided text, select their own additional resources to use, and determine their own solution to the research questions.The math teachers and the librarian have together selected an anchor text about city planning and provided support for students as they work deeply into the text, using it as evidence to support their analysis.

### Part VII: Summative Assessment Description and Rubric

Using a framework of city planning guidelines students will plan a city based on fractal design. Students will write a justification of the fractal pattern they used and the placement of required city elements on their plans. Students will use text-based evidence to support their decisions, placements, and plans. Using mathematical terminology, construction tools, and algebraic equations students will justify placement of city elements and focal points based on city planning guidelines. Students will answer two complex questions and pose several of their own for future research.

Students will use this City Planning Proposal Guide.

Summative Rubric

### Part VIII: Prior Knowledge Needed

An understanding of fractions, the golden ratio, symmetry, and basic geometric shapes.

How to support claims using text-based evidence.

A basic understanding of how cities function based on human needs.

### Part IX: Student Learning Objectives

1. The student will be able to use fractals, geometric shapes, and similarity transformations to map a city of their own design.
2. The student will be able to apply geometric methods to solve problems by finding creative solutions to physical constraints on a map.
3. The student will be able to answer research questions and broaden the inquiry by using textual evidence from a complex anchor text.
4. The students will be able to solve city planning problems by analyzing and reflecting on the anchor text.

### Part XII: Attachment of Student Work Examples

Student Samples:

Completed Day 2 Lesson

Completed Summative Planning Document

Completed Summative Planning Document

Completed Summative Proposal

Completed Fractal Drawing

Completed Unit Reflection from Student

### Part XIII: Teacher and Librarian Reflection on the Implementation of the Lesson

Librarian reflection: The students were engaged in the lesson from day one to day 6 (and we implemented during the last 6 days of school). I found it fascinating to see that nearly all of our students chose a different fractal on which to model their cities. Between two full classes, we had 10 different designs.

On anchor text day we found a nice balance on asking questions to the class. We had never worked together before, so it was interesting to note how one of the math teachers asked more questions of the students, while the other sat in her class and participated as if she were a student. Both models worked great and encourage student participation and engagement.

It was fun for me to check in with the classes after the anchor text day and watch the students working on their summative project. I took a lot of pictures and posted them to Twitter and Instagram and got a lot of great feedback from other teachers and administrators at our school as well as from librarians from all over the country.

After we received the student reflections, I wanted to make sure we made a more clear connection between the anchor text and the fractals behind it. I might move the slides about city shapes and their fractal similarities to the same day as the anchor text. I would also find an additional text about city planning.

Math teacher reflection:

The first time you present something, you realize the things you needed to spend more time emphasizing in the rubric. Because we presented this at the end of the school year we essentially ran out of time and had to adapt the rubric because of the lack of time.

The unit would be more complete if a connection with Social Studies teacher was also incorporated.  The students would benefit from understanding city designs and growth over centuries.

An overlaid map of Washington DC today would be a great addition to the unit. Students could see how the final layout compares and contrasts from the original plan. Pictures of the street view of cities with parks and focal points, such as can be found with Google Earth would help to highlight the importance of focal points in the design of their city.

The video for recursive drawing should be watch as a group and the teacher should demonstrate how to use the application. I would also include some student activities where two fractal designs would be drawn intertwined.

Overall the lesson went smooth. Co-teaching was a highlight of the unit as it is always great to work with the Librarian and reflect on our teaching.  The final projects and proposals demonstrated knowledge of fractals and the use of the text to design their city.

My personal growth gained by participating in the STEM ISKME unit was greater knowledge of literacy standards and the implementation of important skills students need to be successful. I also increased my knowledge of Creative Common License and OER.

School Librarians Advancing STEM Learning, Granite State University, Concord, NH, February 2016. Funding provided by IMLS.