Arizona Academic
Standards Addresses:
Subject Area: Mathematics
Strand 3: Patterns, Algebra, and Functions
Students will make sense of the world by discovering, exploring, extending,
and creating patterns to organize information and make predictions; will
represent and describe mathematical relationships in a variety of ways.
Concept 1. Patterns. Identify patterns and apply pattern
recognition to reason mathematically.
• PO 1: Patterns: Communicate grade level appropriate recursive
pattern, using symbols or numbers.
Strand 4: Geometry and Measurement
Students will recognize and describe geometric properties and ideas; interpret
spatial relationships and demonstrate spatial sense.
Concept 1. Geometric Properties. Analyze the attributes
and properties of two and three dimensional shapes and develop mathematical
arguments about their relationships.
• PO 4: Geometric Properties: Classify three dimensional figures
by their attributes.
Subject Area: Science
Standard 2: History and Nature of Science
2SC-E1. Identify major milestones in science that have revolutionized
the thinking of the time.
• PO 1. Describe the effects of major scientific events on society.
2SC-E2. Describe how science and technology are interrelated.
• PO 1. Describe a technological discovery that influences science.
• PO 2. Describe a scientific discovery that influences technology.
2SC-P4. Trace the development and consequences of an invention, theory,
or discovery to demonstrate the dynamic nature of science.
• PO 1. Trace the development of a selected invention from its inception
to modern day.
Subject Area: Technology
Standard 5: Technology Research Tools
Students utilize technology-based research tools to locate and collect
information pertinent to the task, as well as evaluate and analyze information
from a variety of sources.
5T-E1. Locate information from electronic sources.
• PO 1. Identify electronic research resources.
• PO 4. Construct keyword searches including basic Boolean logic
using available electronic research resources.
Learning Objectives:
• Students will be able to create their own Spiral of Archimedes.
• Students will be able to compare the natural spiral to the spiral
of Archimedes.
• Students will be able to list facts about the life and inventions
of Archimedes.
• Students will be able to explain why the Archimedes' Screw is
so helpful.
• Students will be able to locate two web sites containing
information on Archimedes and his discoveries.
• Students will be able to predict and explain what will happen
when the center peg of the spiral has a different diameter.
Materials for this lesson:
• Computer
• Student computers with internet access
• Zip drive
• Speakers
• Projector and screen
• Archimedes’ Spiral iMOVIE
• Archimedes’ Screw PowerPoint
• Model of Archimedes screw (meat grinder)
• “Spiral Maker”
• Hammer
• Hole punch
• Piece of wood
• 12X12 construction paper
• Pencil
• Yard stick
• Scissors
• Colored pencils
• String
• “Discover with Archimedes” worksheet
Instructional Activities:
- Review the Natural Spiral with the Class.
- Where would we find a natural spiral?
- The natural spiral is based on what pattern?
- How can you create this spiral based on the Fibonacci sequence?
- Describe this spiral to someone who could not see.
- Explain How We Will be Studying a Different Spiral Today.
- Ask: "Does anyone remember who Archimedes was?"
- Greek mathematician, physicist, and inventor
- Born around 287, died 212 BC in Syracuse, Sicily
- Father was Phidias, an astronomer
- Discovered density when trying to see if King Hieron's
crown was really made out of gold
- Devised a mathematical exponential system to express extremely
large numbers and claimed he could figure out how many grains of
sand would fit in the universe
- He changed math into technology
- Discussion: Archimedes Invented Many Machines Used as Engines
of War.
- They were used in defense of Syracuse when attacked by Romans.
(During the 2nd Punic War)
- His inventions held off the Romans for over 3 years!
- Catapults that could throw objects that were over 500 lbs.
- He invented poles that would reach over the city walls and drop
heavy stones in the ships
- He invented large hooks attached to levers lifted the ships out
of the water and then dropped them
- The soldiers of Syracuse shine sunlight off special shields and
reflect sunlight through giant lenses to set Roman ships on fire
- Discussion: Archimedes discovered one of his most famous inventions
(Archimedes' Spiral) on his way to Egypt.
- He studied with successors of Euclid in Alexandria.
- He sent statements of his latest theorems, without giving proofs.
Some of the mathematicians claimed them as their own, so the last
time he sent some, two of them were false!
- On his way down to Alexandria on the boat, he came up with this
idea we now know as the Archimedes screw.
- It is a tube bent around a rod into the shape of a spiral.
- To operate the screw, it angles into the water so that the under
portions of the threads at the submerged end can cup and hold the
water. When the screw is rotated on its axis in the proper directions,
the cupped water is lifted up along the underside of the beam on
the moving threads until it reaches and can flow out the tube's
raised end.
- In Roman times, the Archimedes' Spiral was operated by walking
on it, as lumberjacks today walk on and spin a floating log.
- When it came into general use in the 15th century, it was rotated
by turning a crank.
- It is a mechanical water pump and is still used today (to irrigate,
the Hemopump, water treatment plants, etc.)
- Show examples of Archimedes' Screw and how it is used today.
- Show Archimedes’ Screw PowerPoint.
- Archimedes also worked with another spiral.
- He called it a helix and said that it was "already studied
by his friend Conan."
- He explored this curve for the first time around 225 BC.
- It is an arithmetic spiral which increases in radius at a fixed
amount.
- The distance between the wingdings is consistent.
- Explain how it is different from a Natural Spiral.
- Used on ships for keeping time.
- Show "Creating Your Own Spiral of Archimedes" imovie.
- Have the Students Predict What Will Happen When the Diameter
of the Peg is Changed.
- Explain to the class that they will also be expected to finish
the worksheet by doing Research on the internet.
Assessment:
Students will create their own spiral and record findings on the "Discover
with Archimedes" worksheet.
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