Applying Pascal’s Law
Lesson #03
Lesson Title: Applying Pascal’s Law
Author(s):
Les Humphrey
Jim Friedel
Donna Johnson
Renee Parsley
Lesson 2 of 2
Lesson Objective:
Supplies Needed:
Phone Number(s):
302-856-0961
302-856-0961
302-856-0961
302-856-0961
Lesson #03
E-mail Address(es):
lhumphre@groupwise.sussexvt.k12.de.us
jfriedel@groupwise.sussexvt.k12.de.us
djohnson@groupwise.sussexvt.k12.de.us
rparsley@groupwise.sussexvt.k12.de.us
Students will demonstrate a working knowledge of :
Evaluating algebraic expressions using formulas
Product and quotients, finding area of a circle
Application of Pascal’s Law, create a unique solution to an opened-ended technically-related problem
Any type of piston, ruler, calculator
THE "7 ELEMENTS"
TEACHER NOTES
(and answer key)
1. Introduce the CTE lesson.
Larry’s truck continued:
Students will apply Pascal’s law and their knowledge of
We remember that Larry’s truck had a major braking problem that was the braking system to solve the problems incurred by
created when he added the swamper tires. You suggested that replacing Larry’s truck.
brake parts that are bigger could fix the problem. But, what do we need Multiple correct answers will be possible, this enables
to know about the original brake system and how much change would students to proceed in discovering a solution appropriate to
we need to do?
their math ability level.
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Applying Pascal’s Law
Lesson #03
2. Assess students’ math awareness as it relates to the CTE lesson.
Pascal’s Law says that an external pressure applied to a
Does anyone know Pascal’s Law? Give a brief background and fluid in a closed vessel is uniformly transmitted throughout
explanation of the law.
the fluid.
Pascal’s Law is: P = F/A
Does anyone have an idea of what P, F or A represent in this formula?
What could be appropriate units for pressure, force and area?
Pressure = Force/Area
Pressure could be in units like pounds per square inch (psi)
Force in unit like pounds (or kg)
Area is measured in square units.
For our problem, we will use psi, pounds and square
inches.
Do you know any other formulas that are structured like this one—
Value A = Value B divided by Value C?
Some students should be familiar with
(Someone might say T = D/R or R = E/I or some other familiar
formula.)
Distance = Rate  Time
Ask students if it would be OK to rewrite P = F/A as F = PA.
Volts = Amps  Resistance
They should agree that these are equivalent equations.
Then do you know any formulas that look like F = PA?
Circumference = π  Diameter
(This should encourage students to say D = RT or E = IR or C = π d)
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Applying Pascal’s Law
Lesson #03
Pascal’s Law works just like these other formulas that you are familiar
with.
Make the connection here with the algebraic formulas by
showing students how to manipulate symbolically the
variations on Pascal’s Law to solve for Force and then
F =Force
again for Area.
Then show the comparison as compared to the triangle
method displayed on the right (memory device often used
in the shop and in science class).
F
P
P = Pressure
P = F/A
A
A = Area
OR
F = PA
A = F/P
* Vocabulary note - It has been common in the auto field
to refer to the area of the top of the cylinder as “surface
area” However the term surface area is a math term that
represents the sum of all areas of the circle. Refrain from
using surface area when referring to the area of the top of
the cylinder.
3. Work through the math example embedded in the CTE lesson.
Describe the brake system in a car and show how the force applied to
the brake system generates a pressure through the top of the cylinder Refer to diagram supplied in PowerPoint or use a diagram
throughout the brake lines, using the visual diagram. Label the values that you would typically use with your students from their
for force and area of the top of the cylinder, then walk the students text or available resources.
through the calculation steps in using Pascal’s Law.
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Applying Pascal’s Law
Lesson #03
Let’s say the piston in this car has a surface area of .5 in2. We will also
assume a 100 lb person can apply 50 pounds of force to the brakes. How
much pressure will be produced on the cylinder?
P = F/A
Additional examples in powerpoint, using brake systems and master
cylinder examples.
P = 50 pounds/.5 in2 = 100 psi
4. Work through related, contextual math-in-CTE examples.
1. If a force of 100 pounds is applied to a piston in a hydraulic
cylinder of 4 square inches in area, what is the pressure?
2. If we have 50 pounds of force applied to an area .5 square
inches, what would be the pressure?
3. If 80 pounds per square inch of pressure is applied to a 2 square
inch area. What is the total force?
4. If total force is 120 pounds and pressure is 30 pounds per square
inch, what is the area?
The problems to the left are designed to be used after the
presentation or given to the students as an in class
assessment of their contextualized mathematical ability.
These problems could also be pasted into a worksheet
form and given as homework or a quick formal
assessment.
1) 25psi
2) 100psi
3) 160lbs
4) 4 in2
5. A master cylinder has 100 pounds per square inch of pressure
applied to the piston. The diameter of the caliper piston is 3
inches.
A) Find the area of the caliper piston in square inches
B) Determine the force exerted on the caliper piston.
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Applying Pascal’s Law
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5A) Area of the Caliper Piston:
r  1.5in
  3.14
A   r2
A   (1.5in ) 2
A  7.065in 2
5B) Force on the Caliper Piston
F  P* A
pounds
) * 7.065in 2
in 2
F  706.5 pounds
F  100 psi (
6)
6. In the previous example, if the area of the caliper piston on the
23
rear brakes is
square inches, determine the force at the wheel
32
caliper piston.
23 2
in  .7188in 2
32
F  P* A
pounds
) * .7188in 2
2
in
F  71.88 pounds
F  100 psi (
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Applying Pascal’s Law
7. A master cylinder has 100 pounds of pressure applied to a 1
square inch piston. If the diameter of the caliper piston is 3
inches.
A) Find the area of the caliper piston in square inches.
B) Determine the force of the caliper piston.
5. Work through traditional math examples.
1. Find the time traveled for a distance of 150 miles at a rate of
55miles per hour. Remember: T = d/r
2. Find the diameter of a circle that has a circumference of 36 in.
Remember: C =  d
3. Solve for the value of x. 15 = x / 3
4. Find the value of r when 3 = 57/ r
5. Find the value of R in terms of E and I from the formula, E = IR
6. Find the average rate traveled when the distance from point A to
point B was 92 miles and the time of travel was 2 hours and 15
minutes.
7. Find the value of x when 36 = 4 / x2
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Lesson #03
7 A) 7.068 in2
B) 706.8lbs
The problems to the left are designed to be used after the
presentation or given to the students as an in class
assessment of their generic mathematical ability. These
problems could also be pasted into a worksheet form and
given as homework or a quick formal assessment.
Questions are embedded in these additional problems
that are designed to extend the basic level of math
calculations and challenge the basic student. The level of
difficulty progresses as you work through the problems.
1)
2)
3)
4)
5)
6)
7)
2.72 hours
113.04 in
x = 45
r = 19
R=E/I
40.89 mph
X=2/9
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Applying Pascal’s Law
Lesson #03
Use problems created in step 4 as demonstrations on
board/overhead. Ask appropriate questions to check for
Students will complete practice problems using three variations of understanding, ask students to explain steps and solutions.
Pascal’s Law. In both parts 4 and 5.
Students will develop a solution to Larry’s truck problem and submit Refer to PowerPoint activities for classroom instructions
answer in written form providing mathematical support using Pascal’s
Law
6. Students demonstrate their understanding.
7. Formal assessment.
Students will be assessed upon their written, mathematical
submissions to Larry’s truck problem. They will provide a written
solution to the designed problem illustrated at the beginning of the
lesson. Students will be required to provide documented work
and proof of mastery of the mathematical concepts in Pascal’s
Law dealing with “Larry’s Truck” as well as a written report
summary of their mechanics report. Alternatively, students could
complete the standard worksheet that is more directed and can be
used at a lower student ability level than the open response
problem detailed in the powerpoint.
Have students write and describe using focus free writes to tell
you what they learned during the lesson. Included in the writing
should be several examples that they have come up with. The
students should do their focus free write for 4 minutes. (Or
whatever you deem necessary)
FOCUS FREE WRITE- a paragraph writing assignment that
describes what the student has learned throughout the lesson. This
Math-in-CTE: Automotive Technology
Additional Formal assessments can be made to meet
student needs and ability levels such as creating quizzes or
additional practice problems from the problems designed
and listed above.
Worksheets have been developed and will be attached to
this lesson.
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Applying Pascal’s Law
Lesson #03
is an opportunity for students to reflect on the day’s lesson. This
should be done in silence and for designated short (ex.5 minutes)
time, which is whatever is appropriate.
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