6th Grade Supporting Idea:Geometry and Measurement

Teacher Quality Grant

Supporting Idea 4• MA.6.G.4.1: Understand the concept of Pi, know common

estimates of Pi (3.14; 22) 7 and use these values to estimate and calculate the circumference and the area of circles.

• MA.6.G.4.2: Find the perimeters and areas of composite two-dimensional figures, including non-rectangular figures (such as semicircles) using various strategies.

• MA.6.G.4.3: Determine a missing dimension of a plane figure or prism given its area or volume and some of the dimensions, or determine the area or volume given the dimensions.

MA.6.G.4.1

MA.6.G.4.1

Why Do I Have to Learn This?

Because circles are seen almost everywhere!

To determine if a tree is big enough for a car to drive through.

To determine a bicycle’s distance based on the circumference of the wheels.

What is π

• Formal Definition from Wikipedia: – π (sometimes written pi) is a mathematical constant

whose value is the ratio of any circle's circumference to its diameter in Euclidean space.

– This is the same value as the ratio of a circle's area to the square of its radius.

– It is approximately equal to 3.14159265 in the usual decimal notation.

– Many formulae from mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants

What is π

• Formal Definition from Wikipedia: – π is an irrational number, which means that its value

cannot be expressed exactly as a fraction m/n, where m and n are integers.

– Its decimal representation never ends or repeats. – It is also a transcendental number, which implies, among

other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value;

• Proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics.

What is a Transcendental Number• Proving this was a late achievement in

mathematical history and a significant result of 19th century German mathematics.

• A number (possibly a complex number) which is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients.

• Most common are π and e

Euclidean Space

• Formal Definition from Wikipedia:– In mathematics, Euclidean space is the Euclidean

plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions.

– The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.

Euclidean Space Spherical Space

Euclidean Space

• Formal Definition from Wikipedia:– From the modern viewpoint, there is essentially only one

Euclidean space of each dimension.

– In dimension one this is the real line;

– In dimension two it is the Cartesian plane;

– In higher dimensions it is the real coordinate space with three or more real number coordinates.

The History of π

• The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius (π=3), but one Old Babylonian tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.

The History of π

• Ancient Egyptians calculated the area of a circle by the following formula (where d is the diameter of the circle):

– This yields an approximate value of 3.1605 for pi.

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8d

9

⎛

⎝ ⎜

⎞

⎠ ⎟2

The History of π

• The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), he worked out that 223/71 < π < 22/7.

Archimedies• A genius for mathematics with a physical

insight;

• Rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.

• Method of Exhaustion to find π

• The last words attributed to Archimedes are "Do not disturb my circles"

Archimedes

• Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape.

• Archimedes had to find the if there was silver in the kings gold crown and he had to solve the problem without damaging the crown.

• He could not melt it down into a regularly shaped body in order to calculate its density.

Archimedes

• While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown

• Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!”

Archimedes

Prepare a Lesson in 35 Minutes

  History and Mathematics In your group describes how Archimedes discovered pi. Organize a way that you will present this to the other students in class. You will have seven minutes to present this lesson.

http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html

 

The History of π

• Archimedes's approximated the area of a circle by finding the area of a regular polygon inscribed within the circle and the area of a regular polygon circumscribed about the circle.

– GeoGebra 1» Inscribed Worksheets: Triangle, Square & Hexagon» Circumscribed Worksheets: Triangle, Square & Hexagon

– GeoGebra 2 (doubling method)

The History of π

• GeoGebra 3

• This file shows a visual demonstration of unrolling a circle and comparing the perimeter to the Diameter.

• It shows how there are 3 diameters and a little extra in the perimeter.

Lesson 1 - The Ratio of Circumference to Diameter

http://illuminations.nctm.org/LessonDetail.aspx?ID=L573

Lesson 2 - Discovering the Area Formula for Circles

http://illuminations.nctm.org/LessonDetail.aspx?ID=L574

Apple Pi

Materials You Will Need

Lesson 1

Pieces of string, approximately 48" long

Circular objects to be measured Apple pies (or other circular food item, to be measured at the end of the lesson)

Apple Pi activity sheet

Calculators

Rulers

Lesson 2

Circular objects Calculators Scissors Compasses Rulers Area of Circles activity sheet Fraction Circles activity sheet Centimeter grid paper on overhead transparencies Blank copy paper

Eratosthenes

“Explore”Eratosthenes

Observe this 6½ minute video from Youtube. It contains Carl Sagan’s narration of how Eratosthenes measured the circumference of the earth.

“Explore” In Groups of 2, 3 or 4

Please review the lesson below.http://www.ciese.org/curriculum/noonday/

 

Based on the lesson and the video,

Draw a picture of how they might use this experiment in their sixth grade classroom. What type of preparation would they need?

“Investigate” with GeogebraCircumference of a Circle

Dynamic Worksheet

By Michael Powellhttp://www.geogebra.org/en/upload/files/MichaelPowell/Circumference_Investigation.html

Reflection 1

• Complete a Technology Infused Reflection Log based on Michael Powell’s Geogebra unit

“Investigate” with GeogebraArea of a Circle

Dynamic Worksheetby Edward M. Knote High School Math Coach School: Blanche Ely High 1201 NW 6th Avenue Pompano Beach FL 33060 Email Address: Edward.Knote@Browardschools.com

Circle2.zip

http://www.geogebra.org/en/wiki/index.php/User:Knote#Area_Dynamic_Work_Sheets

Reflection 2

• Complete a Technology Infused Reflection Log based on Edward M. Knote’s Geogebra Dynamic Worksheet

“Practice” with the Promethean Board

Review terms and then practice calculating the circumference and the area of a circle using the promethean

board and flip charts.

area_of_a_circle.flpCircumference and Area.flp

ActivInspire ActivInspirePi-day.flp

ActivInspire

Reflection 3

• Complete a Technology Infused Reflection Log based on the ActivInspire

• And Promethean Board Website

Content and Pedagogy Post-test

• You are now ready to take your post-test• Good Luck, and I hope you enjoyed the

lesson

Pi Day

Pi day Activities

• Pi Day is celebrated on March 14, or in the month/day date format as 3/14; since 3, 1 and 4 are the three most significant digits of π.

• March 14 is also the birthday of Albert Einstein and Waclaw Sierpinski so the events are sometimes celebrated together.

• Pi Approximation Day is held on July 22, or in the more common day/month date format as 22/7, which is an approximate value of π.

Pi day Activities

• Exploratorium Pi day

• Education world

• TeachPi.org

• NCTM Pi Day

MA.6.G.4.2

MA.6.G.4.2

MA.6.G.4.2

MA.6.G.4.2

MA.6.G.4.3

MA.6.G.4.3

MA.6.G.4.3