Tension Test
Day One
Mathematics Content/ Connection to Previous Lessons: Students will use data
provided to create linear models. They will create graphs, equations, and tables to
represent the setting. They will also use what they know about the tension test to explain
their formulas and graphs and the relationships between them.
Students Will Be Engaged In: Working in small groups to make sense of the data that
they have been provided with.
Materials: Calculators for all students with the data for the case entered in lists 1 and 2;
worksheets for all student; sample of aluminum to help students understand what is
happening with the test.
TEACHING NOTES
Timetable
15 min
1. LAUNCH: Have students read through the introductory paragraphs on
the student worksheet. Discuss what is happening in the test. Pass
around the sample piece of aluminum so that students can see what is
being tested. Remind students how to deal with data in their
calculators. Make sure all students have the required data.
50 min
2. EXPLORE: Students work in their groups on the questions from the
student worksheet. The goal for the first day is to have students
complete the student worksheet through question 13.
Ask questions such as: How can we decide what window to use when
graphing the data? How can we find an equation for a linear
relationship? How can we figure out what the y-intercept and slope
mean?
Look for: Do students recognize the linear relationships? Are they
able to express these relationships in equations? Are students able to
interpret their results in terms of the test?
The teacher may need to help students to come up with a plan for
finding equations for the linear relationships. Some suggestions might
be using the linear regression function of the calculator, or picking two
points and finding slope and intercept from those.
15 min
3. SHARE & SUMMARIZE: Have the class discuss their strategies for
finding equations for linear relationships. Discuss the ways that the
setup of the test shows up in the equations, graphs and tables.
Copyright 2003 Syracuse University Mathematics Education program
Developed through the Algebra by Design project: A collaborative project between SU and SCSD
Ask questions such as: What does the y-intercept of each of your
equations represent? How can we tell from a graph how long our
original sample was? What is the relationship between change in
length and new length? How does this show up in your graphs, tables
and equations?
Look for: Are students able to relate their mathematical
representations to their understanding of the physical elements of the
test?
4. APPLY: Students will continue working with these ideas during the
next day as they look at stress and strain.
Assessment Activities: Students should be completing the student worksheet as they
proceed with the activity.
Additional Notes to Aid Teachers: Consider a bar of initial length I subjected to force F
at both its ends. The bar has a rectangular cross section. Let A be the area of cross section
of the bar. The force F produces a uniform stretching of the bar, and the bar is said to be
in tension.
m
F
F
I
L
F
F
To know what’s happening inside the bar let’s make an imaginary cut at section m as
shown in the above figure. We now look at the part of the bar to the left of the cut.
F
S
The tensile force F acts at the left end of this cut part of the bar; at the other end are
forces that represent the action of the removed part of the bar on this part. These forces
Copyright 2003 Syracuse University Mathematics Education program
Developed through the Algebra by Design project: A collaborative project between SU and SCSD
are continuously distributed over the cross section. The intensity of force (i.e., force per
unit area) is called the stress and is denoted by S.
F
S
A
When the bar is stretched by the forces F, as shown in the figures above, the resulting
stresses are tensile stresses; if the forces are reversed in direction causing the bar to be
compressed, the resulting stresses are compressive stresses. Stress is expressed in pounds
per square inch (psi) or kilo pounds per square inch (ksi).
The bar undergoes a change in length, becoming longer when in tension and shorter when
in compression. The change in length denoted by L has been shown in the above figure.
Strain is defined as the elongation per unit length and is denoted by E.
E
L
I
If the bar is in tension, the strain is called a tensile strain, representing an elongation of
the material. If the bar is in compression, the strain is a compressive strain and the bar
shortens. Because strain E is the ratio of two lengths, it is a dimensionless quantity; that
is, it has no units and is expressed as a pure number.
Copyright 2003 Syracuse University Mathematics Education program
Developed through the Algebra by Design project: A collaborative project between SU and SCSD
Tension Test
Day Two
Mathematics Content/ Connection to Previous Lessons: Students will use data
provided to create linear models. They will create graphs, equations, and tables to
represent the setting. They will also use what they know about the tension test to explain
their formulas and graphs and the relationships between them.
Students Will Be Engaged In: Working in small groups to make sense of the data that
they have been provided with.
Materials: Calculators for all students with the data for the case entered in lists 1 and 2;
worksheets for all student.
TEACHING NOTES
Timetable
10 min
1. LAUNCH: Recall with students the setup of the test. Have them
discuss what they learned in day one about the linear relationships.
40 min
2. EXPLORE: Students work in their groups on the questions from the
student worksheet. The goal for day two is to have students finish the
student worksheet.
Ask questions such as: How can we decide what window to use when
graphing the data? How can we find an equation for a linear
relationship? How can we figure out what the y-intercept and slope
mean? How do we find the area of a rectangle? How do we find the
units of a ratio?
Look for: Do students recognize the linear relationships? Are they
able to express these relationships in equations? Are students able to
interpret their results in terms of the test? Are students able to explain
what the appropriate units are for different quantities?
10 min
3. SHARE & SUMMARIZE: Have students share their thinking about
units. Discuss the ways that the setup of the test shows up in the
equations, graphs and tables.
Ask questions such as: What does the y-intercept of each of your
equations represent? What does the slope of each of your equations
represent? Why doesn’t strain have any units? Why does stress have
units?
Copyright 2003 Syracuse University Mathematics Education program
Developed through the Algebra by Design project: A collaborative project between SU and SCSD
Look for: Are students able to relate their mathematical
representations to their understanding of the physical elements of the
test? Do students understand the way that units of ratios are found?
20 min
4. APPLY: Engage students in a discussion of Hooke’s Law. Discuss the
relationship between Stress and Strain. A material is said to be elastic
if on removing the applied load or pulling/pushing force it regains its
original shape. Most structural materials are elastic up to a particular
value of the applied pull. If we try to plot the relationship between
stress and strain we will find that the stress-strain curve is a straight
line, which passes through the origin. This means that the stress is
proportional to the strain. This kind of relationship is called a linear
relationship. When a material behaves elastically and also shows a
linear relationship between stress and strain the material is said to be
linearly elastic. The relationship between stress and strain can be
expressed by the equation S  YE in which Y is a constant of
proportionality known as the modulus of elasticity. The units of Y are
the same as the units of stress because strain is dimensionless.
Different materials have different Y. The above equation is commonly
known as the Hooke’s law.
Assessment Activities: Students should complete the student worksheet.
Copyright 2003 Syracuse University Mathematics Education program
Developed through the Algebra by Design project: A collaborative project between SU and SCSD