TIME REQUIRED

Two or three 50-minute lesson periods.

LESSON RATIONALE

Simple mathematics can be used to solve various problems involving gears.  When students know how to find a mathematical solution to these types of problems, they will be able to more fully understand how engineers design such things as bicycles, car transmissions, electric can openers, dishwashers, and blenders.  You and your students can solve problems involving gears the same way engineers do.

LEARNING OBJECTIVES

By the end of this lesson on gears the student will:

1. Define, understand, and demonstrate the concept of gear rations and how it relates to engineering design.
2. Be able to calculate the speed of a secondary gear knowing
   • Primary gear speed
   • Gear diameters
3. Understand that gear ratios may be selected to multiply force at the expense of speed (ratio of primary to secondary less than unity) or to multiply speed at the expense of ease of operation (ratio greater than unity). (You can use this observation as a way of introducing the principle of conservation of energy, by explaining that there is always a trade-off).
4. Be able to apply the following mathematical skills to solve real problems:
   • Multiplication
   • Division
   • Ratios
   • Substitution of values for variables in an equation

MATERIALS REQUIRED

• Copies of handouts for each student (the student reading for this lesson and the Mathematical Glossary).
• Demonstration to represent a gear ratio (a set of gears, a bicycle, a pulley system, etc.).
• Pictures or photos of gears and gear applications.

LESSON CONCEPTS TO BE LEARNED

Engineering Concepts

Gears are wheels with teeth that mesh together.  Motion in the first, or primary, gear causes motion in the opposite direction in the secondary gear.

The speed, or number of revolutions per minute (rpm), of any secondary gear is dependent upon its diameter and the diameter and speed of the primary gear.  The rpm of the secondary gear can be calculated by the following equation:

N_s x d_s = N_p x d_p

N_s = rpm of secondary gear

N_p = rpm of primary gear

d_p = diameter of primary gear

d_s = diameter of secondary gear

The diameters of the gears are given or can be measures, and the speed of the primary gear is usually given.

When N_p is greater than N_s, the speed of the secondary will be faster than that of the primary, but the person or machine driving the primary will have to apply extra force to turn it.  When N_s is greater than N_p, the speed of the secondary will be slower than the speed of the primary, but the primary gear will now be easier to turn and the secondary gear will turn with force greater than the force with which the primary is turned.  With gears, you can always multiply speed or multiply force, but you cannot multiply both at the same time.

Mathematical Concepts

1. Use of mathematical variables to represent actual physical quantities
2. Simple equation solving
3. Substitution of values for variables in an equation
4. Application of ratios and ratio solving

A ration is a comparison of two numbers, expressed as the quotient of the first number divided by the second number.

Formula for ratios: a/b = c/d; also ad = bc; therefore a = bc/d

The student will be asked to use skills in multiplication, division, and subtraction.   Cross multiplication is also used.

RECOMMENDED USE

This lesson can be used to introduce the most basic algebraic skills to younger students, or to build on more advanced students' algebraic skills.

LESSON PLAN

STEP ONE:

The teacher will bring into class a bicycle, a hand drill, an egg beater, and old watch, a toy, and/or a music box that has gears.

STEP TWO:

The teacher will allow the students to operate (where appropriate) and/or see the interaction of the gears in the object brought into the classroom.

Demonstration Alternatives:
Lego Technic II kit 1032 ($45.00)
Bicycle pedal and rear wheel
Quarter, rotating together with a penny
Styrofoam circles of different sizes

STEP THREE:

The teacher will ask the questions:

1. What is the purpose of gears?
2. How do gears work?
3. Can gears make a job quicker?  Can they make it easier? Why?

Acceptable student responses:

1. To transfer motion or power from one moving part to another; to cause something to turn in a place away from where the original force is exerted; to multiply force; to multiply speed.
2. Motion in one gear (the primary) forces another gear (the secondary) to turn in the opposite direction.  The speed of the secondary depends on the diameters of the two gears and on the speed of the primary.
3. The right gear ratios can be used to multiply force or to multiply speed, but never both at the same time

STEP FOUR:

The teacher discusses the idea that the interaction of the gears with respect to redirecting motion and power is based upon ratios.

STEP FIVE:

The teacher distributes Student Reading #1 on ratios and gear ratios, and explains the math formulas for ratios and applies them to the diagram.

LESSON ASSIGNMENT

Students will bring to class an item or object that uses gears.  (This item may utilize multiple gears or gear sets, but the student need only calculate the speed of any two gears.)  Have the students calculate the rotations of both gears (rpm) based upon the diameter measurements.

ENRICHMENT ACTIVITIES

Gears expand the physical limits of the human body.  Have students think of something that could be designed and/or created using gears that would immediately help an elderly grandparent do something that is no longer easy, help a handicapped person, help a small child, or help them when they are sick.  Students should write a short paragraph of explanation and draw pictures of their designs.

Encourage students to develop their observation skills by listing 10 examples of gears used in the home.  Ask them to draw an example of gears in the home or bring a demonstration to school.

Have students go to the library and research gears and their effects on the workplace.