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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
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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
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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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