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StaircasesA 10 th Grade Expedition to Answer theQuestion: Are the Staircases in the Cityof Boston Safe?Grade 10

Expedit ion AuthorKaren M. CrounseCodman Academy Charter Public SchoolBoston, MA

Expeditionary Learning SchoolsLearning Expedition Documentation Pro ect


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Summary

Each year, approximately 25-30 students participate in the Staircases learning expedition, devoting 90minutes per day, four days a week to their intensive and extensive study of staircases in the city of

Boston. The expedition spans two semesters, with the first semester being a whole class curriculum,

and the second semester being primarily independent work.

The theme of this expedition is What makes a staircase safe?. The investigation of this question is

divided into two components: the first, focusing on measuring and observing staircases, making

calculations with data, and graphing, and the second, emphasizing analyzing and reflecting on

mathematical concepts and data related to staircase construction, learning about state building codes,

and redesigning a staircase. The mathematics content includes linear equations, slope, solving andgraphing equations, scale drawings, parallel and perpendicular lines, Pythagorean Theorem, statistics

and one-variable graphs, scatter plots and interpreting and drawing conclusions about data.

Classroom activities focus on understanding the mathematical content, while the project componentsallow practice and application of skills in terms of staircase design and safety.

The staircases expedition involves deep classroom explorations of key state and district mathstandards as well as extensive fieldwork,

community service, expert consultation, and

presentations; these aspects of the expedition

are not typically associated with a high school

math curriculum and they help the materialcome alive for students. Students spend a great

deal of time in their Dorchester neighborhood,

and in other areas of the city, measuringstaircases, meeting with building inspectors and

architects, and working with property owners to

ensure that their staircases are safe. Students

also present their findings to parents andcommunity members.

Guiding Questions

! What is steepness and how can we define it?! How can we compare staircases graphically and algebraically?! What mathematics is needed to understand staircases?! How can data be summarized?! How can data be used as evidence to draw conclusions?! How can we apply our mathematical understanding of staircases to solve similar problems?



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Investigation Topic 1

Collecting and Analyzing Data

The first investigation of this expedition focuses on collecting data, graphing staircases, and usingstatistics to summarize data. Students complete staircase measurements and calculations, and

summarize their data using one-variable and two-variable graphs. Students also complete a project

reflection.

Long-Term Learning Targets

" I can solve everyday problems thatcan be modeled using linear

equations and can solve linear

equations.

" I can use the appropriate graphical,tabular or symbolic methods to

solve problems.

" I can select, create and interpretappropriate graphicalrepresentations for a set of data,

including using appropriate

statistics to communicate

information about a set of data." I can solve simple triangle

problems involving triangle sum,

Pythagorean Theorem or special right triangles.

" I can solve real-world and symbolic problems involving linear functions and their graphs, tablesand equations

" I can draw a line a regression for a set of (x,y) data, determine an appropriate equation andexplain the meaning of the equation.

Investigation Topic 2

Analyzing Data and Staircase Redesign

The second investigation of this expedition focuses on using the data from the first investigation to

answer the question Are the staircases in the city of Boston safe?. The answer to this question takes

the form of a safety report that students present to building owners and public leaders in the city.Components of this investigation include journal questions, student designed questions, MCAS open



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response questions, a summary of building codes, a safety report, a staircase redesign (including

calculations), letters to building owners, and a final project reflection.

Long-Term Learning Targets

" I can demonstrate an understanding of the mathematics of staircases and apply my understandingto solve related problems.

" I can summarize data statistically and graphically and use it as evidence to draw accurateconclusions.

" I can summarize statistical information graphically using stem-and-leaf plots, histograms, boxplots and scatter plots and draw conclusions from the graphical representations

Expeditionary Learning SchoolsLearning Expedition Documentation Project


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Connections to State and District Standards

Math

Patterns, Relations and Algebra" Students will demonstrate an understanding of the relationship between various representations of a

line, determine a lines slope and x- and y-intercepts from its graph or from a linear equation thatrepresents the line, find a linear equation describing a line from a graph or a geometric description ofthe line, e.g., by using the point-slope or slope y-intercept formulas, and explain the significance ofa positive, negative, zero, or undefined slope.

Geometry" Students will solve simple triangle problems using the triangle angle sum property and/or the

Pythagorean Theorem.

Data Analysis and Statistics" Students will select, create, and interpret an appropriate graphical representation (e.g., scatterplot,

table, stem-and-leaf plots, box-and-whisker plots, circle graph, line graph, and line plot) for a set ofdata and use appropriate statistics (e.g., mean, median, range, and mode) to communicate informationabout the data, and use these notions to compare different sets of data.

" Approximate the line of best fit (trend line) given a set of data (e.g. scatter plot)." Use technology where appropriate.

Number Sense" Students will find the approximate value for solutions to problems involving square roots and cube

roots without the use of a calculator, e.g., 8.2132

!" , and students will use estimation to judge thereasonableness of results of computations and of solutions to problems involving real numbers.

Patterns, Relations and Algebra" Students will describe, complete, extend, analyze, generalize, and create a wide variety of patterns,

including iterative, recursive (e.g., Fibonnacci Numbers), linear, quadratic, and exponential functionalrelationships. Students will solve everyday problems that can be modeled using linear, reciprocal,quadratic, or exponential functions, apply appropriate tabular, graphical, or symbolic methods to thesolution, include compound interest, and direct and inverse variation problems, and use technologywhen appropriate.

Geometry" Students will use rectangular coordinates to calculate midpoints of segments, slopes of lines and

segments, and distances between two points, and apply the results to the solutions of problems.Students will find linear equations that represent lines either perpendicular or parallel to a given lineand through a point, e.g., by using the point-slope form of the equation.



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Major Project

This expedition has one major project with ten significant components. The components arediscrete, but interrelated. There are a number of tangible products that emerge from these project

components, some for classroom use only, and some for outside audiences.

1. Staircase Measurements and Calculations

Description

In order to respond to the expedition question, Are the Staircases in the City of Boston Safe?

students collect data about staircases in the City of Boston. Students complete their analysis ofstaircases using a minimum of thirty staircases as follows:

" Twenty class staircases: These staircases are measured during class fieldwork days. They

include a variety of staircase types from staircases around school and the city. This whole-class

activity serves as a model for the process that students use to collect data on their personalstaircases.

" Ten personal staircases: These are staircases measurements that are individual to the student.

They can be from around their home and neighborhood. No other student should have thesestaircases in their data set.

Since the project entails understanding staircases as a whole, students collect data on a variety of

staircases indoor, outdoor, residential, commercial, long, short, steep, less steep. Multiple sectionsand landings are all to be represented in the data. Students complete an accurate sketch of each

staircase showing the number of steps and placement of the landings. Calculations for staircases

include slope, horizontal length, staircase height and diagonal length (using the PythagoreanTheorem).

2. Staircase Graphs with Calculations

Description

To get an accurate picture of their staircase data, students graph their staircases. For their tenpersonal staircases, students select five to create scale drawings. Drawings must be precise and show

the entire staircases. Of the five graphed, at least two staircases must have multiple sections. These

drawings must accurately reflect the measurements and calculations shown on the data collection

forms. Calculations for the staircase graphs include the linear equations for each section and landingwithin the staircases. These must be clearly labeled with the equation, slope and y-intercept. These

graphs allow students to graphically compare different staircases.



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3. Summarizing Data: One-Variable Statistics and Graphs

Description

For students to answer the question,Are the staircases in the City of

Boston Safe? the data need to be

organized and summarized. Using

their data set of thirty staircases,students calculate statistics including

mean, median, mode and the five-

number summary for six variables.

These variables include slope, rise, runand three others (number of steps,

number of landings, number of

sections, length or height) selected by

the student. Since understanding slopeis a primary theme in this expedition,

students create three one-variable

graphs for slope. They create a stem-and-leaf plot, histogram and box plot.

This allows students to see different

representations of the same data as

well as understand their slope data indifferent ways.

Since it is valuable to compare the values of rise and run (especially when learning about the building

codes), students create two graphs that are comparable. That is, for the rise and run data, they createtwo box plots, two histograms or two stem-and-leaf plots that have identical scales so the values can

be compared.

For the remaining three variables, students create one graph for each variable. To give students morepractice at creating and interpreting different graphs, one of these graphs must be a box plot, one must

be a histogram and one must be a stem-and-leaf plot.

Although students write initial observations of the graphs, they are further analyzed in the secondinvestigation.

4. Comparing Variables: Two-Variable Graph and Analysis

Description

Through their work with staircases, students realize that most staircases have a rise that is shorter thanthe run. To examine this relationship further, students create a scatter plot, graphing the (run, rise)



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coordinates of their 30 staircases. Using the scatter plot, they draw a line of best fit and determine the

equation for the relationship. They then use the equation to predict values. At times in this

expedition, this activity was the students first encounter with writing equations for lines of best fit,so the assignment sheet leads them through the process step-by-step. Students have the opportunity

to create other two-variable graphs during the student-designed journal question in the second

investigation

Having students think about whether to create a two variable graph or compare two one variable

graphs is important it helps to rephrase the question in the form of looking for a relationship if

there is one. In some cases, using a two-variable graph and looking at the y=x line can help students

interpret the relationship. Students define their question, develop a hypothesis and create a processneeded to answer it.

5. Journals

Description

The journals allow students to write about their

understanding about slope and staircases using their

data as evidence to support their ideas and

conclusions. Any questions that can guide studentsthrough the process of defending their ideas using

their data are appropriate to use. The process for

responding to the question is detailed and structured

to require students to use their data to drawconclusions. This is the skill that is being

emphasized in the journal writing process. The

format leads students through organizing their data

to help support answers so they will be comfortablewith the process for further questions both in written

and oral form. Students completed journal entries on

the following topics:

" Journal How do Rise and Run Contribute to

Steepness?" Journal How Does Adding a Landing Affect

the Overall Slope of a Staircase?

" Journal How do the Slopes of Indoor andOutdoor Staircases Compare?

" Journal Student Designed Question and Response



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6. MCAS Open Response Question

Description

The state of Massachusetts administers their high school graduation exam in the spring of the 10 th

grade year. Throughout the stairs expedition, students complete practice questions from the stateexam as Do Now and problem set questions. To give students further practice, one open-response

question from a previous MCAS exam is included in the expedition. The rubric is directly from the

state assessment of the question and it is scored on a four-point scale like the actual test. Students

work through this question and revise until they earn a score of 4; they get an understanding of whatan advanced response looks like. This part of the rubric can be for any standard assessment practice

question that requires explanation or work shown.

7. Summary of Building Codes

Description

Throughout the expedition, students have created and refined their own definition of what makes a

staircase safe. By reading the Massachusetts State Building Codes, students learn the definition of

safe as created by the state of Massachusetts. This information is important to answer the guidingquestion about safe staircases in Boston. Students are given the section of the state building codes

that relates to staircases. As part of the project, they are asked to summarize the building codes.

Aspects of staircases discussed include the walking surface, width, landings, nosings, headroom, rise

and run ranges, guardrails and handrails. In addition, there is information about spiral staircases andramps, with their own sets of requirements. To support understanding, the town building inspector is

invited in to present information about the codes.

The Massachusetts state building codes are located on the web at the following address:www.mass.gov/bbrs/code.htm. The following sections focus on staircases: 780 CMR 1014.0 and

780 CMR 3603.9 through 3603.15.3.

8. Safety Report: Evaluating Data

Description

At this point, students have a deep understanding of staircases including the mathematics involved

and they are ready to explore what makes staircases safe. In the Safety Report, students summarize

their information to respond to the Expedition question, Are the Staircases in the City of BostonSafe? Students interpret their data based on whether their staircases meet the building codes or not.

To guide student thinking, questions are provided as a structure. It is important to emphasize that

students must use their data in their responses for example, when determining if their staircases



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meet the codes, they can review their data and specifically state how many dont meet the codes and

for what reasons. Generally, the focus is on the rise and run measurements, since the data collectionprocess did not include all of the variables included in the building codes. The data collection form

could be altered to include more variables if desired.

9. Staircase Redesign and Letter to the Owner

Description

The staircase redesign gives students the opportunity tomake rise and/or run corrections to the staircase they deem

the most hazardous. Students are required to have a scale

drawing of the original staircase and the redesign. In many

cases, students may have already created a scale drawingin the first part of the project. If not, they need to create an

original scale drawing. The redesign needs to meet the

building codes. The most difficult part of this process isensuring that the redesigned staircase meets the original

height and length. In many cases, students can reduce

landing lengths to play with the length; however, the

height of the staircase must reach the precise height of theoriginal. To do this, students can change the number of

steps as well as adjust the rise and the run. The goal is to

make the staircase safer and to meet building codes. Other

additions to the redesign include railings, guardrails, treadmats, and lighting. All additions must be to scale. As with

the other scale drawings, the actual graph must include a

title, labeled axes, scale and the overall length and height.

Note: using an expert architect who can show students howto make true architectural drawings of their redesigned staircases could enhance this part of the

project.

The calculations for the scale drawing are the same as required in the first part of the project. Inaddition to the linear equation for the different sections of the staircase and the landings, students

must write the equation for the railings. This helps to further remind students that parallel lines havethe same slope.

The letter to the owner provides a format for summarizing the changes made to the staircase in the

redesign. Students explain why the staircase is unsafe, outlining why it does not meet the building

codes. Then, they explain the changes they made to the staircases to improve its safety.



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speak to the class; he summarizes information and answers student questions. Finally, students areasked to summarize each section of the related code in their own words.

Writing

There are a variety of writing assignments within the expedition. Students have multiple revisions forall documents, allowing them to create a product of which they are proud. The form for the journal is

modeled before they create a question of their own. In addition, peer edits are frequently used to

allow students to get feedback before work is turned in to the teacher.

Math

Although the expedition is focused on mathematics, there are mathematical skills that are embedded

that are not part of the overall learning targets. Fractions and decimals are necessary and practiced as

part of measuring with a ruler and understanding slopes. In addition, students review calculationsincluding mean, median and mode as well as measuring angles.

Arts

As part of the project, students redesign a staircase that is unsafe and make it safe. Although themathematical accuracy is key, students add features to make their design their own.

Connections to the Community and Larger World

Fieldwork" Measuring staircases in the

school neighborhood andthroughout the city

Service

" Analyzing staircases in theircommunity and identifying

unsafe ones.

" Letters to city representativesand owners of unsafe

staircases.

Experts

" Building Inspector for Understanding Building Codes" Architect for Re-Design of Staircase

Exhibitions

" Formal oral presentation of work for parents, teachers and students.



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Calendar

During the first semester of the year, the class content is aligned with the first part of the

expedition project and components are partially completed in class or as home assignments.During the second part of the project, the majority of the work is done by students outside of

class. There are a few exceptions the meetings with the building inspector and architect and

approximately 6 other class days for students to catch up/get support for their work. These

additional days are especially important toward the end of the project when students may berevising multiple pieces, organizing their binders and preparing for their presentations. Overall,

the expedition is designed to be a full-year mathematics expedition.

Part I: Data Collection and Graphing

September October November December

"StaircaseMeasurements and

Calculations

"Staircase Graphswith Calculations

"Summarizing Data:One-Variable

Statistics and

Graphs"Staircase Graphs

with Calculations

"Summarizing Data:One-Variable

Statistics and

Graphs"Comparing

Variables: Two-

Variable Graph

"Project Reflection

Part II: Analyzing Data and Staircase Redesign

January February March April May

" JournalQuestions

" StudentDesigned

Question

" MCASOpenResponse

Question

" Summarizing Building

Codes

"Safety Report:EvaluatingData

" StaircaseRedesign withCalculations

" Letter to theOwner

" Final ProjectReflection


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