Fractals
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: The 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.
Part IV: Standards Addressed
CCSS Math Standards
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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).
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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
Criteria | Exemplary | Proficient | Basic | Limited |
I have done everything to be in “Proficient” plus: | I have done everything to be in “Basic Proficiency” plus: | I have done everything to be in “Limited Proficiency” plus: | ||
City Map | The city map includes multiple fractal designs. | The city map includes accurate geometric methods, language, and calculations to design the city. + City is mapped using similarity transformations in fractal designs. + The map clearly illustrates focal points. | All components of the city are included. + City is mapped demonstrating fractal design. | Some of the components of the city are included. + City is mapped using geometric shapes. |
City Planning Proposal | The proposal uses multiple references as evidence to justify the plan. + The proposal explains, using similarity transformations and the meaning of similarity between figures in placement of key city characteristics.. + The proposal broadens the inquiry process by including additional questions for further research.. | The proposal uses text-based evidence to support the analysis of the city map and the use of fractals to solve a design problem. + The proposal demonstrates an understanding of city design and answers all questions citing evidence from the text. | The proposal justifies key placement of focal points and cites specific evidence from the text. | The proposal attempts to answer questions |
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
- The student will be able to use fractals, geometric shapes, and similarity transformations to map a city of their own design.
- The student will be able to apply geometric methods to solve problems by finding creative solutions to physical constraints on a map.
- The student will be able to answer research questions and broaden the inquiry by using textual evidence from a complex anchor text.
- The students will be able to solve city planning problems by analyzing and reflecting on the anchor text.
Part X: Text Set Description
Text Title & Hyperlink | Text Purpose | Text-Dependent Questions (created by the teacher/librarian to help students analyze the text) | Accommodations for Diverse Learners |
THE PLAN OF [link to full text] | This is our Anchor Text, designed to provide content while provoking student inquiry around the essential question. The Quantitative text complexity of the text is: 11.3 | 1. What question did the author seek to answer? 2. How do well-planned cities grow? 3. What is his primary claim? 4. Did he provide specific and useful evidence? 5. The word “fractal” is not mentioned in the text at all. How is it implied? 6. Which of the fractal shapes would best fit L’Enfant’s design? 7. What geometrical shapes make up a beautiful city? 8. Why did L’Enfant study other city plans prior to designing D.C.? 9. When are focal points used in translations, dilations, and reflections? 10. How did L’Enfant use focal points? 11. How do we plan for a city to grow? 12. Where do you see evidence of fractals in L’Enfant’s map? 13. Which types of fractals do you see on L’Enfant’s map? | 1. Define complex vocabulary for all students. 2. Specific chunks have been chosen to support students in breaking the reading down into manageable sections. On this version we numbered the pages and adjusted the font and white space on the original document to aid in student understanding. |
Supporting Text Mathigon: Fractals & Dimensions [link to full text] | This is our supporting text, designed to help students understand the mathematic functions of fractal geometry. | 1.How does this reading connect to the anchor text? 2. After reading the text, get together as a small group (3-4) and ask as many questions as you can. Write down the questions exactly as they are asked without judgement. 3. Look over your questions and categorize your questions into open-ended vs. closed-questions (closed questions are where the answer is yes or no, or a right or wrong answer). Change any closed-questions into open-ended questions. 4. Share your top three questions with the rest of the class. |
Part XI: Suggested Lesson Breakdown/Pacing
Day (80 minute periods) | Student Learning Objectives | Aligned Student Learning Task and Suggested Timing | Formative Assessment | Important Accommodations |
Day 1 | DISCOVERY: TSWBAT identify the basics of fractal design. | Students learn what fractals are by watching videos, (more videos) reading articles, viewing images, asking questions, and discovering fractals on their own. | Exit Ticket: one attempt at asking questions and creating a fractal using one of the simple rules in this Google doc. | The librarian will provide the videos and selected article ahead of time for students with high need. |
Day 2 | Student Exploration: TSWBAT create their first fractal drawing. Librarian/Math-Teacher Co-taught Discussion: TSWBAT understand the connection between fractals, geometry, and cities. |
| Exit Ticket: one attempt at drawing a fractal using Recursive Drawing website and answers to the questions. | |
Day 3 | Librarian Led Lesson: ANCHOR TEXT: TSWBAT understand the connection between Geometry and L’Enfant’s city planning process by reading and annotating the |
| 1. The librarian and math teacher will monitor progress to see if certain areas of the assignment are confusing. The teacher and librarian will collect the handout and give feedback before the next class period. 2. The librarian will choose annotations to share aloud, tracking trends and noticing gaps or misconceptions. | 1. The librarian will provide a list of defined vocabulary for the student to use during the 2nd reading. 2. The librarian will provide the article ahead of time to students with high need. 3. The librarian will guide students in need to the page of the text containing the actual answers. 4. The librarian can select some of the questions (not all) for some students to answer. |
Days 4-6 | SUMMATIVE: TSWBAT begin planning their own city using Geometry and fractal design. | Students will have up to 3 days, while working in pairs, to create their cities using the City Planning Proposal Guidelines, and understanding of fractals, and the use of Geometry in city planning. [Make a copy of the City Planning Proposal sheet in order to share with your students] | The librarian and math teacher will monitor progress to check for understanding. | The math teacher will combine students based on interest and needs. |
Additional Day(s)? | Creativity and Research: TSWBAT begin to research an aspect of the summative that especially interested them. | The librarian will lead students in asking their own questions about fractals, geometry, city planning and begin mapping out a short research plan to be completed on their own time (or if time allows). | Students complete a Google reflection on the unit. |
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
Reflection on Student Work (from Pam and Becky):
We found that the student work we received reflected one major problem with our unit: We didn't give our students enough time to complete their summative assessments. We watched them have fun while they worked together on creating their own fractal cities, but everyone was frustrated by the extreme time constraints. If we could do it again, we would give our students an extra week of time to complete their fractal cities. We added much of the high level inquiry (students generating their own questions) to the end of the unit and now that we reflect on the unit as a whole, we could have incorporated more high level inquiry throughout the entire plan.
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.