The Legend of Paul Bunyan

Unit Plan for a

Mathematical Journey

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Karen L. Beard

Lynda Lombardi

Course: EDU 464

Theme-Based Unit Plan

Prof. Hari Koirala

October 8, 2003


The Legend of Paul Bunyan

Unit Plan for a Mathematical Journey

 

Table of Contents

 

 

Unit Plan Overview

 

1.  Unifying Theme................................................................................................ 2

2.  Unit Assumptions............................................................................................. 2

3.  Grade Level(s)................................................................................................. 2

4.  Unit Topics....................................................................................................... 2

5.  Classroom Resources.................................................................................... 2

6.  Unit Features................................................................................................... 3

7.  Unit Objectives................................................................................................. 3

8.  Tentative Unit Timeline.................................................................................... 4

9.  Concept Map.................................................................................................... 5

10.  Unit Plan Connections and Extension

a. Lessons within the math unit.................................................................... 6

b. Unit in the context of other math concepts and units................................ 8

c. Unit extension into other subject areas..................................................... 9

11.  Unit Alignment with State and National Standards

a. Content...................................................................................................... 9

b. Process................................................................................................... 10

12.  Assessment................................................................................................. 11

 

 

Lesson Plan

..... 13.  Lesson Plan #3.......................................................................................... 13

..... 14.  Lesson Plan #3 Activity Sheet.................................................................... 16

..... 15.  Lesson Plan #3 Homework Sheet............................................................. 17

..... 16.  Lesson Plan #3 Quiz.................................................................................. 18

                                 

 

Appendix

A.  The Legend of Paul Bunyan Key Facts............................................................ 19

  1. Unit Resources................................................................................................ 19

 

 


The Legend of Paul Bunyan

A Mathematical Journey

 

1.  Unifying Theme

The theme for this unit is the tall tale The Legend of Paul Bunyan.  The unit will be introduced to the students by reading them a book of Paul Bunyan’s adventures, and the lessons within the unit will explore mathematical concepts based on the facts and inferences from the book, as well as other sources on Paul Bunyan tales. 

 

 

2.  Unit Assumptions:

It is assumed that the students will have a general, and not an in-depth, understanding of the following concepts: 

·         Scale, ratio, and proportion

·         Measurement in two and three dimensions (area and volume)

·         Decimal, fractions, and percentages

 

The unit is designed to provide a deeper understanding of these concepts, as well as to introduce other topics through hands-on activities, independent and group work, thought provoking questions and problems, and the usage of multiple mediums, including literary works, reference books, maps, modeling clay, and the internet.

 

 

3.  Grades Level(s):

This unit has been designed for grade 7 or 8 audiences, although it can be modified to teach grades 2 – 12.   (See Section # 10, page 8-9, for potential adaptations for lower or higher level audiences).

 

 

4.  Unit Topics

 Topics to be covered in The Legend of Paul Bunyan unit are as follows:

  • Measurement
    • Estimation
    • Standard and non-standard units of measure
    • One, two, and three dimensional measurement (length, area, volume)
  • Decimals, Fractions and Percentages
  • Data presentation and Graphing
    • Tables
    • Bar diagrams
    • Pie charts
    • Linear graphs
  • Scale, Ratio, and Proportion
  • Variables and Linear equations
  • Probability and Statistics
  • Data Analysis and Problem Solving

 

 

5.  Classroom Resources

 

The following resources are needed to teach this unit:  The Legend of Paul Bunyan book for unit introduction and reference, calculator with graphing ability, U.S. Map, modeling clay, rulers, yard sticks, graph paper, atlas, and computer with access to the internet.

 


6. Unit Features 

Important features of this unit include:   

·         Mental visualization and estimation

·         Modeling for exploring mathematical concepts

·         Cooperative learning

·         Use of technology, including calculators, computers, and internet

·         Data analysis and problem solving

·         Multiple modes of assessment, including independent work, group work, group projects, student assessment of other student/groups assignments, homework assignments, teacher observation, quizzes, and unit test.

·         Linkages to other concepts, topics, and subject areas (See Section 10, pages 6-9)

 

 

7.  Unit Objectives

 

At the end of this unit the student will be able to:

  1. Estimate and explore using standard and nonstandard units of measure.
  2. Build a cardboard replica of Paul Bunyan’s ‘axe handle’ to use as a unit of measure.
  3. Identify the advantages and disadvantages of using nonstandard units of measure
  4. Create 3-dimensional clay models for Paul Bunyan’s ox, Paul Bunyan’s frying pan, and Lake Superior to extend understanding of the concepts of length, area, and volume.
  5. Determine scale factors for two-dimensional and three-dimensional models.
  6. Set-up and solve ratio and proportion equations, based on their models’ attributes, to determine actual dimensional qualities (height, width, area, volume, etc) of objects and places from the Paul Bunyan tall tale.
  7. Demonstrate and explain how square units of measure can be used to measure objects that are not “square”.
  8. Conduct research using atlases, books, and the internet to gather data on lakes within Minnesota (including surface area, volume, and location) and other states in the U.S.
  9. Represent data pictorially through the creation of bar graphs, pie charts, and tables.
  10. Convert data from decimals to fractions and percentages.
  11. Analyze data to solve related word problems.
  12. Color-code a map of the United States to express ranges of water-land ratio percentages for each state.   Report observations.
  13. Utilize data and maps to identify patterns and consistencies with the legend of Paul Bunyan and U.S. topography to prove or disprove the validity of the legend’s tales.
  14. Create line graphs from numerical data tables 
  15. Convert line graphs to linear equations. (Students will be introduced to both the point-slope equation and the slope-intercept form.)
  16. Calculate the specific slope of the line.
  17.  Find the mean, mode, median of data from the folk tale and data they have researched, including data from Minnesota lake survey data on fish.
  18. Calculate the probability of catching certain types of fish in certain lakes and compare the data across other lakes.
  19. Complete a cooperative team project in which students must identify the most effective means of graphically representing data and create word problems related to their data.

 

8.  Tentative Timeline

 

The Legend of Paul Bunyan unit is comprised of 12 math lessons, some of which may take two days to complete.  The unit will take between 3 and 4 weeks to teach, assuming a school class length of 40-50 minutes.  Cross-curriculum topic planning and execution by teachers within the grade is encouraged; it would be ideal to have English/Literature and Social Studies teachers incorporate topics pertaining to story telling, folk tales, U.S. geography and/or U.S topographical points of interest into their lessons while the math unit is being taught. 

 

The following is a breakdown of the unit’s lessons.  More detailed lesson descriptions can be found in Section 10, pages 6-7.

 

    Lesson      Instructional

    Number          Days        Lesson

        1                  1           Standard and Non-Standard Unit of Measure

        2                  2           One dimensional Measurement and application of scale modeling and proportions

        3                  2           Two dimensional Measurement and application of scale modeling and proportions

        4                  2           Three dimensional Measurement and application of scale modeling and proportions

        5                  1           Graphing (bar graphs and pie charts)

        6                  1           Percentages

        7                  1           Decimals, fractions, and percentages

        8                  1           Graphing points and introduction of linear equations

        9                  2           Continued introduction of Linear equations, variables, slope

       10                 2           Mean, mode, and median

       11                 1           Probability

       12                 2           Problem Solving (and group presentations)


9. Concept Map of

      Unit Mathematics Content     


10.  Unit Plan Connections and Extensions

 

A.  Math lesson sequence within The Legend of Paul Bunyan math unit

 

This unit is comprised of 12 lessons, some of which will require 2 days of instructional time.  The unit will be introduced by reading a book of Paul Bunyan’s adventures.  The book and numerous tales of Paul’s adventures are a source of storytelling exaggerations and numerical references.  Each lesson will begin with a reference to a Paul Bunyan tale which will lead to the exploration of a mathematical concept.  (Note:  Appendix A, Page 19, contains a listing of Paul Bunyan facts which can be used for this unit.)   The following is a brief description of the lessons within this unit: 

 

Lesson #1.  Standard and Non-Standard Unit of Measure.

Lesson involves estimation and exploring standard and non standard units of measure.  In Paul Bunyan’s tales, Paul is reported to be a height of 63 ax handles tall, and his blue ox Babe is 42 ax handles wide from the tip of one horn tip to the other.  Student groups create replicas of axe handles from cardboard and ‘measure’ the room, desks, etc.  Have groups compare findings.  As the lesson progresses, interject that some accounts in another Paul Bunyan legend reference Babe’s width between his horns as 7 axe handles.  Discuss with the class; perhaps this is due to 1 of Paul’s ax handles being equal to the length of 6 ordinary ax handles.   Students will identify advantages and disadvantages of nonstandard unit of measure

 

Lesson #2 – 4.  Measurement and the application of scale modeling and proportions. 

These lessons involve the usage of models for investigating one dimensional (length), two-dimensional (area), and three dimensional (volume) measurements through three-dimensional models and the utilization of scale factors and proportionality.  In these lessons, students will create clay models for Paul Bunyan’s ox (Lesson #2), Paul Bunyan’s frying pan (Lesson #3), and Lake Superior (Lesson #4) where Paul Bunyan is said to have herded whales.  Measurements of the models will be used to set up proportions to identify attributes of things mentioned in Paul Bunyan’s tales.  Note:  A lesson plan for Lesson #3 is included in Section 13, pages 13 – 16.

 

Lesson #5.  Graphing (bar graphs and pie charts). 

This lesson involves having the students use bar graphs and pie charges to represent the volume of various lakes in Minnesota.  Paul Bunyan and Babe are said to have create the 10,000 lakes of Minnesota while they trotted around the state.  Their footprints made impressions in the land, and these indents filled up with water to form the many lakes in the state.  Students will use atlases, books, or the internet to identify lakes for their analysis.  Of the lakes they selected, they would represent the lake volume in a single pie chart and bar graphs.

 

Lesson #6.  Percentages

As a continuation of the looking at how Paul and Babe may have altered the topography of the land, students will be given a map of the United States.  They are instructed to identify states that Paul Bunyan is said to have visited and mark them on the map of the United States. (Some of the “documented” places in the Paul Bunyan tales include Minnesota, Wisconsin, Arizona, Washington, Maine), and mark them on the map of the United States.  Students will also identify states where there is no indication that Paul Bunyan visited.  Students will research what the ratio of land and water is for those states, and then calculate the percentages of water versus land.  Data, such as statistics on acreage of land and acreage of water, can be found in atlases or the internet.  The data the students will be recorded in a table with multiple columns.  The first several columns are labeled Name of State, Did Paul Bunyan Visit?,  Total acreage of the state, Acreage of Water, Acreage of Land, % of Water, % of Land.  There will be additional columns on the sheet which will be used in the next lesson.


Lesson #7.  Decimals, fractions, and percentages

This lesson involves the using the data from the Lesson #6.  Students will add labels to the additional columns from the worksheet in Lesson #6 for converting the percentages of land and water to decimals and fractions.   Students will begin to analyze the data to draw conclusions.  Do the states that Paul Bunyan and Babe visited appear to have higher ratios of water?  How does Minnesota’s ratio of water versus land compare to other states he visited?  (After all, Paul and Babe spent most of their time in Minnesota and, in their travels, “created” 10,000 lakes in the state.)   The teacher will then lead the class in a discussion on how could we modify a map of the United States to help with analyzing our water versus land ratios?  For example, color code different colors or shades of colors by ranges of land or water percentages. Shading states with low %’s of brown, students are given a new map and a complete list of water versus land percentages of all states.  They are asked to color code their maps (or modify by other means they’d like) and asked to report 5-10 facts on their analysis of water/land trends.

 

Lesson #8.  Graphing points and introduction of linear equations

These lessons involve the creation of data tables based on the “factual information’ from Paul Bunyan’s tales. Ideally the class will be divided into groups.   A few ‘facts’ include:  “Saw filers called ‘saw dentists’ could usually file one saw in 30 minutes, and sharpened approximately 20 files a day”,  “Babe the blue ox could eat 30 bales of hay – wires and all – a in a single day”,  “Paul trained giant ants that weighed 2,000 pounds each.  The ants could each do the work of 50 ordinary men.”  In the “ant” fact example, the data in a table would be 1 ant = 50 men, 2 ants = 100 men, 3 ants = 150 men, and so on.  This data would be translated into coordinates, and then plotted on a table with the # men on one axis of a graph and the # of ants on another axis.  The teacher will lead a class discussion to identify similarities and differences between the graphs, asking the students such questions as were they able to use the same unit of measure for each graph?  How did they have to adjust the interval of the X-axis and Y-axis data to suit the data they were graphically displaying?

 

Lesson #9.  Continued exploration of linear equations and slope

The data and graphs produced in Lesson #8 will be used for introducing the idea of representing linear equations as equations with X and Y values.  Similarly, the teacher will use this data to introduce the concept of slope. The teacher will show the students how to graphically display this data and equations on a graphing calculator.   Each group of students will be asked to create one tall tale of their own that involves rate of change. They should then represent data from their tale in a table format and graphing coordinates, and then graph the data.

 

Lesson #10.  Mean, mode, and median

This lesson involves exploring the mean, mode and median.  Students will research the fish-life found in the lakes of Minnesota through lake surveys which include data on number of fish netted, type of fish netted, length of fish netted, weight of fish netted, etc.  This information can be supplied to the students in hand-outs or they may research their own data on the internet.  The Minnesota Department of Natural Resources has a website that provides detailed lake survey for many of Minnesota’s lakes. (http://www.dnr.state.mn.us/lakefind/index.html)  The information will be used to explore the mean, mode and median.  For example, what is the mean length of a white perch netted?  What is the median weight of a smallmouth bass netted?  What is the modal number of fish netted?

 

Lesson #11.  Probability

This lesson involves using the data gathered in Lesson #10 to explore probability.  The data may be used to explore probabilities within one particular lake, or comparing the data across multiple lakes.  For example, what is the probability of catching a brown bullhead in Red Lake?  What is the probability of catching a fish greater than 5 pounds in Round Lake?  Would we have a higher probability of catching a fish greater than 20 inches in length in Leech Lake or Lake Vermilion?

 

Lesson #12.  Data Representation and Problem Solving (and group project and presentations)

This lesson involves having the students, in groups, working to identify the best way to graphically display the data they collected on one lake.  For example, they may choose to display data through pie-charts, bar graphs, line graphs, etc.  The goal of their graphical representation is to allow the audience to quickly discern information regarding their data.  Students will then create 5 word problems based on their data and graphs.  The teacher will collect the problem solving projects from each group and make copies of all projects to re-distribute to the class in a “Problem-Solving Packet”.  The students will solve the word problems of their peers based on the data and graphical displays and then assess the methods of displaying the data and the ease with which the data display assisted with solving the problems. 

 

 

B.  The Legend of Paul Bunyan Unit in the context of other Mathematical Concepts and Units

 

This mathematical unit can easily be extended to other math concepts and also used as the basis to introduce or cover topics not already included in the unit.  Here are a few examples:

 

  1. Geometry:  Lesson #3 on creating a model of Paul Bunyan’s frying pan can easily be extended into the area of geometry and covering in depth topics such as radius, diameter, perimeter, areas of cylinders, etc.  What is the perimeter of Paul Bunyan’s frying pan?  How much pancake batter could the pan hold if it was filled to the top?

 

  1. Calculus and Slope Concept:  Lessons #8 and #9 can easily be used to deeply explore the concepts of linear equations, variables, slope.  Concepts can be taught using graphing paper and simultaneously taught on a graphing calculator.  For example, students can create a line graph describing the number of bales of hay that Babe could eat in one week.  Select two points on the graph, and express as points and .  How do you find slope?  .  Create a linear equation to describe the line on your graph in  format.

 

  1. Exponential, Scientific, and Calculator notation:  Lessons on ways to represent very small or very large numbers.  In terms of the Paul Bunyan unit, students can use lake survey data, which includes lake volume estimates, the number of fish in a lake, etc.  and express in terms of scientific notation.

 

  1. Trigonometry:  Paul Bunyan is said to have created a number of mountains as he created lakes and inlets along the west coast, including Mt. Rainer and Mt. Baker when he created the Puget Sound in Washington.  If you were standing one mile from Mt. Baker measures 10,775 feet in height.  You estimate that the angle from you to the top of the mountain is 30%.  Approximately how far are you standing from the tip of the mountain?

 

  1. Discrete Math:  Students can look for patterns in objects in the Paul Bunyan tales. For example, logs from the trees were stacked as shown below.  How could the students identify how many logs are in the stack without counting?  Is there a pattern?  Can we create a formula to represent the pattern?  How many logs will be in a pile if the stacks are 10 logs high?

 

 

Note on Unit Adaptations for Diverse Learners:   The Legend of Paul Bunyan unit can be adapted for the full range of grade levels and for a wide range of student abilities.  As described above, the unit can easily be extended into other mathematical concepts and topics including as the study of Geometry, Calculus, Trigonometry and Discrete Math.

 

Additionally, The Legend of Paul Bunyan unit can be adapted to suite curriculum standards starting as early as Grade 2.  For example, the lessons could be planned around simple counting, mathematical operations (addition, subtraction, multiplication and division), estimation of measurements, estimation of numbers in the context of the Paul Bunyan legend, basics of Geometry though study of shapes, representing and interpreting data in simple tables or graphs.  For classes at Grade 4 - 6 levels, lessons and projects may be planned with lessons that require higher order mathematical skills, knowledge, and more refined execution of process skills (i.e. problem solving, communication, representation). For example, according to the tales, Paul Bunyan’s dinner menu consisted of seventy pounds of fried potatoes, forty-five pounds of T-bone steak, sixty pounds of ham, sixteen large loaves of bread, thirteen dozen eggs, six hundred and seventy-two pancakes topped with two gallons of maple syrup, and ten gallons of strong black coffee.  Students can plan a similar meal for Paul Bunyan based on proportions or estimate the cost of preparing a dinner for Paul Bunyan based on the prices of a local grocery store. 

 

For an example of how specific lessons may be adapted for usage, see the sample lesson plan on pages 13 – 16.   Page 15 contains a myriad of ideas on how the particular lesson may be adapted for diverse learners.  

 

 

C. The Legend of Paul Bunyan Unit extension into other subject areas

 

·         Social Studies

ü      Ecology and the Environment:  The unit can be extended to the studying the impacts of man on our environment (i.e. logging, pollution), conservation ‘limited’ resources such as forests, and the history and practices of the logging industry such as clear cutting and the regeneration of forest groves.

 

ü      U.S. Geography:  The unit can be extended to the studying natural points of interest, particularly including those 'created' by Paul Bunyan such as Puget Sound, Round Lake in Minnesota, Great Lakes, Grand Titon, Grand Canyon, Mt. Baker, and Mt. Rainer.

 

ü      Culture:  The unit can be extended to the studying the origins and cultural aspects of folktales in the U.S. and other countries throughout the world.

 

·         Science

ü      Geology:  The unit can be extended to the studying the creation of the natural points of interest in the U.S., such as the impacts of glaciers, volcanoes, flooding. 

 

·         English

ü      Students can write their own tall tales and explore the folktales associated with Connecticut (Connecticut Yankee) and regionally (Ethan Allen, Legend of Sleepy Hollow, etc.)

 

·         Music:  The unit can be extended to the study of folk music and historical significance of folk music stories, lyrics, colloquial use of language, etc.

 

 

 

 

11.  Unit Alignment with State and National Standards

The Legend of Paul Bunyan unit addresses and encourages the exploration of many aspects of the National and State content and process standard areas.

 

 

Content Standards

The mathematical concepts within The Legend of Paul Bunyan unit, as outlined in the lesson plan descriptions in Section 10, cover a number of topics outlined within the Connecticut and the NCTM national standards.  Alignment of the unit lessons is as follows:

 

        

Lesson #

Lesson Name

NCTM National Standard

Connecticut Standard

1

Standard and Non-Standard Unit of Measure

 

-  Number and Operations

- Measurement

- Estimation (Rounding and place value)

- Measurement (standard and non standard unit of measure)

2

One-dimensional Measurement and application of scale modeling and proportions

 

-  Number and Operations

- Measurement (length)

- Ratio, Proportions and percents (scale factors, proportions, modeling)

- Measurement (length)

- Number Sense (unit conversion)

3

Two-dimensional Measurement and application of scale modeling and proportions

 

-  Number and Operations

- Measurement (area)

- Ratio, Proportions and percents (scale factors, proportions, modeling)

- Measurement (area)

- Number Sense (unit conversion)

4

Three-dimensional Measurement and application of scale modeling and proportions

 

-  Number and Operations

- Measurement (volume)

- Ratio, Proportions and percents (scale factors, proportions, modeling)

- Measurement (volume)

- Number Sense (unit conversion)

5

Graphing (bar graphs and pie charts)

- Data Analysis and Probability

- Probability and Statistics (tables, graphs and charts)

6

Percentages

 

- Number and Operations

- Number Sense (percentages)

- Patterns

7

Decimals, fractions, and percentages

- Number and Operations

- Number Sense (percentages, decimals and fractions)

- Patterns

8

Graphing points and introduction of linear equations

- Algebra

Algebra and Functions (linear equations)

9

Continued introduction of  Linear equations, variables, slope

- Algebra

Algebra and Functions (linear equations, variables and slope)

10

Mean, mode, and median

- Data Analysis and Probability

Probability and Statistics (mean, median, and mode)

11

Probability

- Data Analysis and - Probability

Probability and Statistics (determine probability of events)

12

Data Representation and Problem Solving (and group presentations)

- Data Analysis and Probability

- Problem Solving

Probability and Statistics (graphs, tables, charts, data organization and analysis)

 

 

Process Standards

 

The NTCM Process Standards, which are embedded in the CT Standards, are addressed and encouraged throughout The Legend of Paul Bunyan.  Students are provided with numerous opportunities to solve problems, develop reasoning and Proofs, and develop communication skills throughout the unit.

 

Unit problem solving

This unit provides opportunities for students to solve problems in a range of formats, including traditional types of question-and-answer word problems to less traditional means through modeling, group projects, and independent data collection and analysis.  Students are not taught just how to ‘solve’ a problem, but how to identify what data is needed and how to obtain the data (through research) and how to analyze and/or display the data so it can be used to solve the problems.  For example, the lessons on ‘mean, median, and mode’ and ‘probability’ rely heavily on the students’ ability to obtain, represent, and draw conclusions on the data.  Additionally, the unit blends independent and cooperative learning.  For example, the first lesson on ‘standard and non-standard units of measure’ encourages teamwork of pairs of students to create their own non-standard unit of measure (Paul Bunyan’s axe handle) which they will then use as part of the lesson activities.  There are other lessons, such as the conversion of data from decimals to fractions and to percentages, which may be presented with students working independently.   The unit project, lesson #12, brings problem solving to an even higher level.  Students must work in groups to determine the best way in which to represent data and then create word problems for their peers to solve.  Because the project will be evaluated by peers, as well as the teacher, the students must strive to have a full understanding of their data, the way in which they are presenting the data, and the wording of their ‘word problems’ which other students will be asked to solve.

 

Develop mathematical reasoning and proofs

Teachers may formally and informally explore reasoning and proofs with students as part of this unit through creating tasks and problems that require mathematical reasoning to investigate relationships.  For example, during the lesson on percentages (Lesson #6), students will have an opportunity to make conjectures on the topography of the states that Paul Bunyan has visited versus the states he has not visited.  They will then gather data to investigate their conjectures through analyzing the water-versus-land ratio of the U.S. states. (Note: Paul Bunyan and his ox are said to have created the 10,000 lakes of Minnesota with his footsteps. If this is the case, can we make a conjecture about the other states he visited?)  They will be asked to provide explanations for their reasoning and support with mathematical data.  When studying modeling and scale factors, the teacher may have students explore the effect of increasing the radius their model by 1 inch intervals, or by doubling the radius multiple times.  What happens to the area?  How can this be expressed as a formula?

 

Communication skills

Communication skills are a vital part of the unit.  As described in the section on Problem Solving, outlines how students will work independently, in collaborative pairs, and in cooperative groups at different points in the unit.  Communication skills will be essential, and it is the teacher’s responsibility to encourage communication not just in small group settings, but amongst the whole class.  Students should be challenged to think and reason mathematically, as well as, to express their thoughts in a clear, coherent and organized fashion - both in written language and orally.  For example, in the lessons on modeling and using scale factors, students should be encouraged to share their thoughts on the mathematical methods by which they can use a small scale models to determine the relative size of Paul Bunyan’s ox or frying pan from the tall tale.  Open class discussions on the mathematical reasoning and rationale should be welcoming to all students of various levels and abilities.  Also, the teacher should encourage the students to discuss and use multiple strategies to solve problems or approach a task.  For example, during the lessons on data representation, students can discuss the many methods in which data can be displayed and what method (bar graphs, pie charts, tables, etc.) may be best given their particular task or data set.  Have students support their reasoning with specific examples.   Encourage students formulate conjectures, and reflect upon their own understanding and the understanding of others. 

 

 

12.    Assessment

 

Assessment will occur throughout the unit via a number of traditional and non-traditional means, including self-assessment, teacher-assessment, and student assessment.

 

As part of Lesson #1, on estimation of standard and non-standard units of measure, the students will self assess their accuracy.  As the lesson activity progresses, students will have an opportunity to adjust their strategy to estimate distances and increase their accuracy. 

 

In the sample Lesson Plan #3 on pages 13 - 16, the lesson is accompanied by a classroom activity sheet and also a homework sheet.  The activity sheet is intended to provide structure to the lesson, reinforce key objectives and guide students through the lesson activities; it is not intended as a stand alone worksheet to be collected and graded at the end of the class.  The teacher should continually move about the class, observe the student groups as they work, and ask probing questions to check for understanding.  Therefore, the activity sheet provides an additional means of assessment, not a stand alone means of assessing student understanding during the class.  Also, by monitoring student responses, the teacher may identify where class discussion is needed to ensure there is a solid understanding of main ideas.  

While the activity sheet is completed by the students in pairs, the homework should be completed independently.  The objective of the homework sheet is to provide reinforcement of the concepts of the lesson and also provide the teacher a means of assessing areas how successful the lesson was and his/her teaching was in meeting the lesson objectives. 

 

The student projects created during Lesson #12 will be assessed by other student groups through the completion of an “evaluation sheet”, as well as the teacher.   All students will have received a copy of the evaluation sheet prior to the start of their work, so they are clear on the assessment structure and expectations.

 

A part of the overall assessment of the unit objectives, the teacher will give the students quizzes on main content area (i.e. data representation through graphs and tables; conversion of decimals to percents and fractions, etc.), and will also give a comprehensive test at the end of the unit.   A sample quiz, based on Lesson #3, is included on page 18.  The students will also be responsible for saving their work associated with the unit in order to compile a portfolio upon which they will reflect on their understanding of key concepts.


13.  Sample Lesson Plan

 

Lesson  # 3

 

Two dimensional Measurement and application of scale modeling

“The One-Acre Frying Pan”

 

 

Objectives

Students will:

1.   Demonstrate and explain how square units of measure can be used to measure objects that are not ‘square’, such as circles.

2.   Create a three dimensional model of Paul Bunyan’s frying pan and use the model to calculate attributes (radius and diameter) of Paul Bunyan’s frying pan through scale factors and proportions.

 

Resources

Modeling clay, ruler, calculator

 

Background

This is the third lesson in The Legend of Paul Bunyan unit plan. The unit was introduced by reading a tall tale book on Paul Bunyan to the class.  Note:  In the second lesson of the unit, students created models to explore one dimensional measurement such as length and width through scales and proportion.  This third lesson builds upon their experience, to use modeling to explore area, a two dimensional measurement, through scales and proportions.

 

Procedure

 

The teacher will re-tell the tale of Paul Bunyan’s frying pan to begin the lesson.  The legend states that Paul Bunyan had a huge appetite, and the cook needed to prepare large portions of food to feed him.  The cook had the blacksmith make a cast iron frying pan that covered an acre of land.  It was so big that he had to have up to 50 men strap strips of bacon onto their feet as skates and skate around the pan’s surface just to grease it.

 

The teacher will facilitate a class discussion on the size of Paul’s frying pan and how we might be able to approximate its actual size, by asking some of the following questions:  “How big is one acre?” “Does anyone know how much land their home property is?”  “Can someone draw a representation of what an ‘acre’ might look like on the chalkboard?”  Discuss how a representation of the acre on the chalkboard or on paper or a model can be at a scale that allows us to explore dimensions of the actual object/item.   “Does an acre have to be a certain shape…can it be square, or round, or oval, or rectangular?”  The teacher tells the students that an acre is 43,560 square feet, and leads discussion on the unit of ‘square feet’ to measure area.

 

The students are asked to work with a partner in the subsequent activities, and an activity sheet will be distributed (see Sample Activity Sheet on page 16).  Depending on class ability, the activity sheet can be modified.

 

1)      Each student is instructed to draw, on a piece of paper, a model of an acre in a square form.  Based on the number of square feet in an acre, the students must calculate and label their drawing dimensions.  (Students will need their calculators to determine the approximate length of each square is 208.71 feet)  When the teams are done with their work, one groups will present their findings and how they calculated the length of the sides of the square to the class

 

2)      The students will make a three dimensional model to explore area.  As part of this exercise, the students will complete a worksheet (see sample activity sheet, page 16).  Students are given modeling clay and asked to create a rectangular prism with the base being a square with an area of ¼ square feet.  Their resulting square will measure ½ foot by ½ foot.  The students are instructed to reshape their prism to a cylinder, being careful to not change the thickness of their model.  Students will be prompted to draw the conclusion that the circle, the top surface of the prism, has the same area of the square, which is ¼ foot.

 

3)      In the next step of the lesson, the students will calculate the approximate radius of Paul Bunyan’s frying pan.  First, they will measure the radius of their frying pan model, and then they will set up a proportion to determine the radius of Paul Bunyan’s frying pan by setting up a proportion.  The three key pieces of information the students will use to set up the proportion are:  a) the measured radius of their model is approximately 3.5 inches, or 0.29 feet.    b)  the known area of their model is ¼ sq feet.   c)  the known area of Paul Bunyan’s frying pan is 43,650 square feet. 

 

4)      The students will use the formula for the area of a circle to calculate the actual radius of a circle with the area of 43,560 square feet.  The teacher will ask the different pairs of students to give their approximations for the radius of Paul Bunyan’s frying pan.  The students will be asked how they can find the actual radius of a circle with an area of 43,560 sq. ft.?   The teacher will introduce the formula for the area of a circle on the chalkboard and ask the students to use their calculators to calculate the actual radius based on the formula. 

 

The lesson concludes with a class discussion on the accuracy of their models and the usage of models to investigate the relationships through scales and proportionality. 

 

 

Lesson Notes

 

This lesson involves important mathematical concepts of unit conversion, scales, proportionality and attributes of circles.  The lesson may easily be expanded into studies of geometry, such as the study of a circles radius, diameter, perimeter, etc, and through the exploration of other shapes. 

 

Depending on the ability of the class and the pairs of students that are working in teams, the teacher could present this lesson as a teacher lead activity or allow the student/student pairs to work more independently with guidance and discussion as needed.  If the latter method is employed, it is recommended that the class debrief on their findings of the activity and discuss the accuracy of models and their use to investigate mathematical relationships. 

 

An important feature of this lesson is the students’ assessment throughout the lesson.  The teacher is engaged with the class to ask questions and lead discussions which will aid in solution finding.  Also, the teacher will circulate amongst the student pairs to answer questions and coach, and will stop the class at points in the lesson to share findings and facilitate discussion.  For example, the students may have difficulty setting up proportions with square units. 

 

The equation               is incorrect. 

       

The equation           is correct.

 

To help the students understand the error in setting up a proportion with different units of measure, the teacher may lead a discussion around a simpler example.  If there are two squares, which measure 9 sq. feet and 16 sq. ft., what are the lengths of the two sides?  Can this be proven with a ratio?  Why not?  These types of questions may assist the students with identifying the need to set up proportions with the same unit of measure, which they can apply to their proportion in the frying pan radius activity.

 

This lesson will prepare the students for the next lesson, in which students will apply modeling and proportionality to volume.

 

Lesson adaptations for diversity of learners

 

For lower level functioning classes, or lower grades levels, the lesson may be adapted to explore the circular models, estimation, and measurement.  For example, the teacher may bring in a real frying pan for the students to have students explore finding the pan’s center and diameter, radius, perimeter, etc.   The teacher may have the students create their own ‘frying pan’ out of modeling clay, which can be used to investigate the attributes of a circle.  Where is the center of the circle?  How did you find the center?  What is the measurement of the radius, diameter, and perimeter of your frying pan?  The students can then work in pairs, combine the modeling clay, and create a larger frying pan, in which they can answer the same questions.  Finally, they might work in teams of 4 and create a large frying pan.  These exercises will show the progression of increasing the size of the circle and its effect on the radius, diameter, and perimeter of a circle.  Depending on the functioning level of the class, the teacher may want to present the concept of proportionality and discuss the relative size of Paul Bunyan’s frying pan, perhaps even having a discussion which leads to walking the students through the calculation of the radius of Paul’s frying pan; Alternatively, the teacher may choose to focus on the concept of measurement and estimation.   What will the diameter be if we combine the modeling clay of 4 students?  How about 8?  What is the perimeter of the clock on the wall?  What is the perimeter of your desk?  Choose an item and identify other items in the classroom that you think may have the same perimeter?

 

For higher functioning students, the teacher may introduce geometry software, where students can explore creating their frying pan models on the computer, numerically describing their model’s center point, radius, diameter, area, and perimeter.  Additionally, the teacher may use this as an opportunity to introduce how the information can be graphically displayed on a graphing calculator.

 

Lesson Assessment:

The activity sheet that will be completed with this lesson will not be graded by the teacher.  The students, as they work in pairs and through the lesson discussions, will be responsible for completing the activity sheet and ensuring its accuracy.   As such, students will constantly be self assessing their performance and knowledge of the concepts throughout the class and will have ample time to ask questions or probe areas of interest.  During the lesson, the teacher will be circulating amongst the students as they work on their assignment to identify students’ strengths and weaknesses, taking time to stop the class or student groups to explain concepts or assist students with their thinking process.   The students will receive a homework sheet to reinforce the concepts covered in class.  The homework will be collected the following day for teacher assessment and grading.  (See homework sheet on page 17.)


* * * *   SAMPLE  IN-CLASS ACTIVITY SHEET: LESSON # 3  * * * *

 


The Legend of Paul Bunyan

Name

Activity Sheet:  The One-Acre Frying Pan

Date:

 

 

Objective:

Paul Bunyan’s frying pan had a cooking area of one acre.

This activity investigates how big a one acre frying pan is and what it may have looked like.

 

1.      On the basis of the reading of Paul Bunyan’s legend, what is the area of Paul Bunyan’s frying pan in square feet?

 

2.      On a piece of paper, draw a model of an acre in the form of a square.  From the information in question 1, calculate and label the appropriate dimensions of your model.

 

3.      Suppose that Paul’s frying pan was a round skillet.  Comparing the dimensions of a round (circular) surface when the area is given in square units can often lead to mistakes.  Building a physical model of the skillet can help us understand the change in dimensions.  The following steps will allow you to investigate and compare the area of the square skillet to the round skillet.

 

a.   Using clay, build a rectangular model (with a square face) with the height equal to ¼ inch and each side of the face equal to ½ foot.  Show that the area of the face has an area of ¼ square foot.  On a piece of paper, draw a picture of your square model and label.

 

 

b.   How many of these squares will ‘fit’ into Paul’s frying pan?  Show how you arrived at your answer.

 

 

c.    Without changing the thickness of your clay model, transform your model from a square face to a circular face.  How many of these circles “fit” into Paul’s frying pan?  How did you come to this conclusion?

 

 

d.   Using a ruler, measure the approximate length of the radius for your clay model.  What is the radius?

 

 

e.    On the basis of your measurement, what is the radius of Paul’s skillet?  Discuss the method by which you found your answer?

 

 

f.     Based on a circle with the area equal to ¼ foot, what would the actual radius for such a circle?      (Hint: )  How accurate was your circular model?

 

 

g.   Using the correct radius from a circle with an area of ¼ foot, find the actual radius for Paul’s skillet.


* * * *   SAMPLE  HOMEWORK SHEET: LESSON #  3  * * * *

 

 

 


The Legend of Paul Bunyan

Name

Lesson:  The One-Acre Frying Pan

Date:

Homework Assignment

 

 

 

1.      Today’s lesson showed how square units, such as sq. ft.,  can be used to measure the area of objects that are square and the area of objects that are not square, such as circles.  Explain how we did this. 

 

 

 

 

 

2.      Paul Bunyan loved pancakes!   Sometimes it would take the cook one hour to make enough pancakes for Paul’s breakfast.  The cook usually prepared a large stack of pancakes that were each ½ acre in surface area. 

a.      What is the surface area of a pancake in square feet?

 

 

 

 


b.      What is the radius of each pancake?

 

 

3.      Find a frying pan or pot in your kitchen that could be used to make pancakes. 

a.      What is the radius of the pan in inches and in feet?

 

 


b.      Using proportions, as we did in class today, find the surface area of the pan.

 

 

 


c.       Using the formula for the area of a circle,, find the area of your pan.

 

 

 

d.      Are your answers to questions 3b and 3c the same?  Explain why?

 

 

 

4.      One day after breakfast, Paul decided to have a small snack.  He asked the cook to make 4 pancakes that each had a radius of 65 feet.  Was the one-acre frying pan large enough for the cook to make the pancakes at the same time?  Write a brief explanation and show your work.

 

 

 


* * * *   SAMPLE  QUIZ,   LESSON # 3  * * * *

 

 


The Legend of Paul Bunyan

Name

Lesson:  The One-Acre Frying Pan

Date:

Quiz

 

 

 

1. Paul Bunyan and Babe his blue ox visited a rectangular house.  The front of the house was 450 feet in length.  One of the sides of the house was 400 feet in length.

 a.   Draw a ‘model’ of the house and label.

 

 

 

 

 

b.   Find the area of the house.

 

 

 

 

 

2.       One day, Paul Bunyan’s cook prepared two pancakes and told Paul that he could only have one of them for breakfast. One pancake was round and had a 2,400 inch radius.  The other pancake was square and had sides that were 1,300 inches in length.   Paul is very hungry and wants to eat the pancake that is the largest portion.  Which pancake would he select and why?  Show all your work.

 

 

 

 

 

 

 

 

 

 

3.      Set up a ratio and proportion (like the one we did in class) to solve the following problem. 

Paul Bunyan brought Babe to a circular pond to get a drink of water.  If the pond had a surface area of 50,000 square feet, what is the radius of the pond?

(Remember:  Our class model had a radius of 0.29 ft. and had an area of 1/4 sq. ft.)

 


 

APPENDIX

 

A.  The Legend of Paul Bunyan Key for usage in Unit: 

There are many tales and legends of Paul Bunyan as retold by numerous authors, each providing a wealth of facts regarding Paul Bunyan’s life and accomplishments.  The following is a listing of ‘facts’ as compiled from numerous resources (see Unit Reference and Resources section below), which can be used as part of The Legend of Paul Bunyan unit:

 

·         Paul Bunyan was 63 ax handles tall.

·         Babe, Paul Bunyan’s blue ox, was 42 ax handles wide from the tip of one horn to the tip of the other horn.

·         Paul Bunyan had a frying pan that covered an area of one acre, which was used to make pancakes.

·         Paul Bunyan and Babe created the 10,000 lakes of Minnesota.  Their footsteps created impressions in the land that filled with rainwater, forming lakes throughout the state.

·         Paul Bunyan trained giant 2,000 pound ants.  Each ant could each do the work of 50 men.

·         Paul Bunyan herded whales in Lake Superior.

·         Paul Bunyan created the Puget Sound in Washington by digging a hole along the west coast of the state, and simultaneously created Mt. Rainer and Mt. Baker.

·         Babe could eat 30 bales of hay, wires and all, in a day.

·         It took a crow a day to fly from one Babe’s horn tips to the other.

·         The legends of Paul Bunyan incorporate a number of actual locations and points of interest in the United States, including:  Maine, Minnesota, Wisconsin, Arizona, Washington, Grand Canyon, Grand Teton, Puget Sound, The Great Lakes, and the Grand Canyon.

 

B.     Unit Resources

 

Buhl, D., Oursland, M.,Finco, K. (2003).  The Legend of Paul Bunyan An Exploration in Measurement.  Mathematics Teaching in the Middle School Focus Issue:  Proportional Reasoning, 8 (8), 441-448.

 

Connecticut State Board of Education (1999).  A Guide to K-12 Program Development in Mathematics.  Hartford, CT: Retrieved September 30, 2003 from http://

     www.state.ct.us/sde/dtl/curriculum_pulb_guide1.htm

 

De Leeuw, A. (1968).  Paul Bunyan and His Blue Ox. Champaign, Ill: Garrard Publishing Company.

 

Eicholz, R. & O’Daffer, P. ( 1993).  Addison- Wesley Mathematics Grade 7.   New York: Addison- Wesley- Publishing Company.

 

Microsoft Encarta. “Bunyan, Paul.” 2000. encarta.msn.com 

 

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics.  Reston, VA: Retrieved on September 30, 2003 from http://standards.nctm.org/document/chapter 2/index.htm.