Home            Forums            Grades Pre K-2            Grades 3-5            Grades 6-8            Grades 9-12

 

 

Instructional Plan Template

Mathematics Governor’s Institute 2006

(Download as Microsoft Word Document or Images: Instructional Plan, Classical Face p1 – Image, Classical Face p2 – Image, Classical Face p3 – Image, Classical Face p4 – Image, Classical Face p5 – Image)

 

Names of group members:

 

  • Marge Ayers: St. Catherine of Siena School, Horsham, PA
  • Joyce E. Cannone: Sharpsville Area Middle School, Sharpsville, PA
  • Terry DiGiovine: Old Forge Jr/Sr High School, Old Forge School, PA
  • Beatrice Ginsburg: Abington Jr. High School, Abington, PA
  • Lawrence Lee: Abington Jr. High School, Abington, PA

 

Topic/Theme: Golden Ratio

 

Level: Grade 6-8 Geometry

 

Time Element: Two 50-minute class periods

 

NCTM Standards Addressed:

 

  • Geometry
    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    • Use visualization, spatial reasoning, and geometric modeling to solve problems
  • Measurement
    • Understand measurable attributes of objects and the units, systems, and processes of measurement
    • Apply appropriate techniques, tools, and formulas to determine measurements

 

PA Math Standards Addressed:

 

  • 2.3.8.A. Develop formulas and procedures for determining measurements (e.g., area, volume, distance).

 

  • 2.3.8.D. Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.

 

  • 2.9.8.J. Analyze geometric patterns (e.g., tessellations, sequences of shapes) and develop descriptions of the patterns.

 

  • 2.9.8.K. Analyze objects to determine whether they illustrate tessellations, symmetry, congruence, similarity and scale.

Math Assessment Anchors Addressed:

 

  • M6.B.2.2  Solve problems involving length, perimeter, area and/or volume of geometric figures.

 

  • M6.B.2  Apply appropriate techniques, tools and formulas to determine measurements.

 

  • M6.C.1.2  Represent and/or use concepts and relationships of lines and line segments.

 

  • M7.B.2.1  Develop, use and/or describe strategies to find the measure of length, perimeter, circumference, area or volume.

 

  • M7.C.1.2  Identify congruence and/or similarity in polygons.

 

  • M7.C.3.1  Locate, plot, and/or describe points on a coordinate plane.

 

  • M8.C.1  Analyze characteristics and properties of two- and three- dimensional geometric shapes and demonstrate understanding of geometric relationships.

 

 

Reading Assessment Anchors Addressed:

 

  • 1.1.8. B: Identify and use common organizational structures and graphic features to comprehend information.

 

  • 1.1.8. F: Understand the meaning of and apply key vocabulary across the various subject areas.

 

  • 1.6.8. A: Listen to others.
    • Ask probing questions.
    • Analyze information, ideas, and opinions to determine relevancy.
    • Take notes when needed.

 

  • 1.6.8. D: Contribute to discussions.

o       Ask relevant, probing questions.

o       Respond with relevant information, ideas or reasons in support of opinions expressed.

o       Listen to and acknowledge the contributions of others.

o       Adjust tone and involvement to encourage equitable participation.

o       Clarify, illustrate or expand on a response when asked.

o       Present support for opinions.

o       Paraphrase and summarize, when prompted.

 

  • 1.6.8. E: Participate in small and large group discussions and representations.
    • Initiate everyday conversation.
    • Select a topic and present an oral reading.
    • Organize and participate in informal debates.

 

 

 

Objectives:

 

  • The students will be able to identify Golden Rectangles.

 

  • The students will be able to recognize the Golden Ratio within art, architecture, and nature.

 

  • The students will be able to identify, continue, and create a Fibonacci sequence.

 

 

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

 

Preclass:      Lead short discussion on who the “hot” celebrities are.  Ask students who the “hot” male and female celebrities.  Limit criteria on visually appealing faces.  

 

Discuss either as a class or as a journal assignment what makes these peoples’ faces visually appealing.  Share responses.

 

Activity:

  • Pass out rulers and pre-made portraits of popular celebrities.
  • Hand out instruction sheet that asks students to make specific measurements.
  • Students are to follow the activity sheet.

            On activity sheet, students will make specific facial measurements and

calculate ratios.

  • When students finish, gather results and compile on overhead.

            A number close to 1.618 should appear often.

  • Pass out worksheet entitled “A Classical Face” for students to work on if they finish early.
  • Discuss the number 1.618, define the golden ratio.
  • Present the history of the Golden Ratio, i.e. appearance in structures and sculptures created by the ancient Greeks.
  • Make connection between Golden Ratio and the concept of beauty linking the mathematics to the opening activity.

 

Closure:

  • Pass out worksheet “Are Golden Ratios in Your Face?”

Students will complete in pairs.

  • Discuss how Golden Ratios are part of everyone – connection to self –image, etc.

 

 

 

Materials/Resources:

 

 

 

 

Interdisciplinary Connections:

·         Reading

o       Research on Fibonacci Sequence and Golden Ratio

 

·         Technology

o       Use of Geometer’s Sketchpad to construct Golden Rectangles

o       Graphing Calculator for scatterplot

o       Online Research

 

·         Other

o       History:  Greek architecture and sculpture (i.e. Parthenon)

o       Art: (ex. Leonardo da Vinci)

o       Science: (Botany)

 

Assessment Strategies:

·         Formative Evaluation (checking student understanding during the lesson):

o        Observation

o        Pair/Share

 

·         Summative Evaluation (How will it be determined that the objectives were achieved?):

o        Summarizers

o        Quiz

 

Correctives/Remediation:

o        Teacher can provide students with pre-made rectangles that students measure to see if the golden ratio is present. 

o        Students can then make their own rectangles on grid paper and measure their rectangles to see if the golden ratio is present.

 

Extensions/Enrichment:

·        Fibonacci Activity

·        Construction Activity

·        Properties Activity

 

Special Accommodations (special needs students)

·          Description of the Special Needs student selected:

This male student is dyslexic, and he struggles with reading and writing.  He is also clumsy and has coordination problems.  However, his mathematic skills are outstanding.

 

·          Accommodations to use with this student:

o       Give special education students any information that needs to be read in advance

o       Auditory instructions should be given to the entire class.

o       Students should work with partners when reading and/or writing (Furthermore, this student can be partnered up with someone who is weak in mathematics because of his exceptional mathematical abilities.) 

o       Chunk assignment into small pieces

o       Give mini-deadlines so students can turn in each piece of the large project separately.

o       Teacher and students should use graphic organizers

o       Give special education students simpler manipulatives to use (i.e. larger calculators)

o       Establish nonverbal cues with special education students. 

o       Monitor students’ progress and provide frequent feedback.

o       Provide checklist

o       Demonstrate and model each part of lesson

Appendix:

 

·        Fibonacci Sequence Worksheet

·        Fibonacci Table Worksheet

·        Fibonacci Scatterplot Worksheet

·        A Classical Face

·        Are Golden Ratios in Your Face?

·        Golden Ratio Worksheet: Practice Measuring

·        Constructing a Golden Rectangle

·        Special Properties of the Golden Ratio

·        Golden Ratio Quiz

·        Differentiated Quiz: Golden Ratio

 

 

Name: _____________________________  Date: ______

 

Period: _____

 

Find the next three numbers in the sequence.

 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ____, ____, _____, …

 

 

Explain how you continued the pattern.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Name: _____________________________  Date: ______

 

Period: _____

 

Surprise !

 (in the Fibonacci Sequence)

 

Take the ratio of the two successive numbers in the Fibonacci’s series and divide each by the number before it.

 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 . . .

 

Complete the table.

Ratio #

Ratios

Decimal (Round to nearest thousandth)

1

1/1

1

2

2/1

2

3

3/2

1.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Name: _____________________________  Date: ______

 

Period: _____

 

Create a scatter plot with the ratios found in your table.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use your graphing calculator to create a scatter plot.

 

What is happening to the value of the ratios ?  Explain.

 

 

 

 

 

 

Constructing a Golden Rectangle

 

1.         Open Geometer’s Sketchpad.               2.         A window will pop up.

In the Edit menu, choose Preferences.                         Click on the Text tab and put a check

                                                                                                in the option for For All New Points

 

Construct a square

3.         Construct segment .                                   4.         Select points A and B, and the segment.

In the Construct menu.

choose Perpendicular Lines.

 

 

 

 

 

 

 

 

 

5.         Select A then B.

            In the Construct menu, choose Circle by Center and Point.

            Construct the point where the circle intersects the perpendicular line through A.

 

6.         Construct a line perpendicular               7.         Hide the lines and construct segments

to line  through point C.                                          to create square ABDC.

 

                                      

 

8.         Select segment .                                         9.         Select E then D.

In the Construct menu,                                                In the Construct menu,

choose Midpoint.                                                         choose Circle by Center and Point.

                               

 

10.       Select points A, then B.

            In the Construct menu, choose Line.

            Mark the point where the line intersects the circle.

 

11.       Hide the circle, and construct a line perpendicular to the line through point H.  Construct a line perpendicular to side  through point C.

Mark the intersection point of the two new perpendicular lines.

 

12.       Hide the perpendicular lines and use segments to construct the rectangle below.

 

13.       Measure AC and AH to verify that ACHI is a Golden Rectangle.

Name_________________________________         Date_____________________

 

Teacher____________________

 

Special Properties of the Golden Ratio

 

The Golden Ratio is represented by the Greek letter f.  f » 1.618.

 

Follow the steps below to find some interesting properties of f.

 

In your calculator calculate .  This will give you an approximation of f.

 

Store this number in your calculator.

 

1.         Subtract 1 from f.                                                        ________________________

           

 

2.         Divide 1 by f.                                                              ________________________

 

 

3.         Add 1 to f.                                                                  ________________________

 

 

4.         Square f.                                                                     ________________________

 

Look back at your results.

 

Which expressions are equal to each other?

 

A.        f - 1                B.         f + 1                C.        f2                     D.       

 

 

 

 

 

 

 

Name________________________________                       Date___________________

 

Teacher___________________

 

Golden Ratio Worksheet:          Practice Measuring

 

In each of the rectangles given, measure AB and BC.  Then find the ratio of the larger number to the smaller number.

 

1

2

3

4


 

5

6

7

8

9

 

 

 

 

 

 

Name________________________________                       Date___________________

 

Teacher___________________

 

Quiz:  Golden Ratio

 

Write the word YES inside the rectangles that are Golden Rectangles.

 

1

2

3

4

5

6

7

8

9

 

 

 

 

 

 

Name: _________________________  Date: ___________

 

 

Golden Ratio Quiz: Multiple Choice

 

 

1.   When one subtracts a square from one side of a golden rectangle, the remaining area has the same proportions as

 

a.   Half the original rectangle

b.    1/3 of the original rectangle

c.    A square

d.    The original rectangle

 

 

2.   The Golden Ratio is

 

a.   Phi

b.    1.618…

c.    The ratio of the dimensions of the Golden Rectangle

d.    All of the above

 

 

3.   To create a Golden Spiral, you start with

 

a.   A circle

b.    A Golden Rectangle

c.    A square

d.    A straight line

 

 

 

 

 

 

 

 

Home            Forums            Grades Pre K-2            Grades 3-5            Grades 6-8            Grades 9-12