Students will   consider how gradient (slope) affects the velocity of an object rolling down   an incline

Burgos, Wilson

Haag, John

Zuniga, Joseph

English

The College at Brockport, State University of New York: Digital Commons @Brockport

The College at Brockport: State University of New York
Digital Commons @Brockport
Lesson Plans CMST Institute
8-11-2004
Connection Between Gradient and Slope
Joseph Zuniga
The College at Brockport
John Haag
The College at Brockport
Wilson Burgos
The College at Brockport
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Repository Citation
Zuniga, Joseph; Haag, John; and Burgos, Wilson, "Connection Between Gradient and Slope" (2004). Lesson Plans. 15.
http://digitalcommons.brockport.edu/cmst_lessonplans/15
 1 
 
 
Names: Joe Zuniga, John Haag, Wilson Burgos 
Grade level(s)/Subject taught:  Earth Science / Secondary Mathematics  
Objectives: Students will use Interactive Physics and TI Interactive Software to: 
• consider how gradient (slope) affects the velocity of an object rolling down 
an incline. 
 
• make a table of values and a graph using the data generated by the 
model. 
 
• calculate and compare the linear and quadratic regressions of the data 
generated by the model. 
 
• consider the influence that a change in one variable has on another. 
 
• make predictions and draw conclusions based on the data generated by 
the model. 
 
• compare data generated by the model graphically, on a table of values, 
and in an equation. 
 
 
Mathematical - Science Concepts or “key ideas” that modeling will be used to teach: 
 
Students will: 
 
• make a connection between gradient (Earth Science) and slope (math). 
 
• use concepts of algebra to write and evaluate algebraic expressions, to look at the 
three views of a function, and to consider linear and quadratic relationships. 
 
• use concepts of modeling and multiple representation to display the data, make 
predictions, and draw conclusions. 
 
• use mathematical reasoning to analyze mathematical situations, make 
conjectures, gather evidence, and construct an argument. 
 
• use technology to help manipulate parameters and apply the mathematics to real 
world situations. 
 
 2 
 
 
 
Essential Questions:  How does spacing of the contour lines on a topographical map affect 
the gradient?  How does a change in gradient (slope) affect the velocity of an object rolling 
down an incline?  What type of function can best model the relationship between slope and 
velocity? 
 
Earth Science  
 
     In Earth Science, students have previously been taught the skills of using a topographic 
map to determine elevation from the contour lines, and measuring distance using the map 
scale. I will point out to the students that this is a map of a small island in the ocean (see 
attached map), so the first contour line would be 0 meters in elevation (sea level). The 
student’s knowledge of these skills will be assessed with a bell work assignment that will have 
them measure the distance between points A and B, and the elevation at each point. Each 
student will have a copy of the map to work from, and I will walk around the room checking 
results. I will then use the overhead to go over the answers to make sure that everyone has 
the same values to begin with (there is an acceptable range of answers for the distance). 
 
     I will then introduce the concept of gradient by asking the students to find the formula for 
gradient in their Earth Science Reference Tables. The students work along with me as we 
calculate the gradient between points A and B, using the values that we found in the bell work 
assignment. I will emphasize that it is important to include the units in the equation, as well as 
the answer. The Interactive Physics program requires us to use the same units for the X and Y 
axis. This would not usually be the case with problems that the students would do in Regents 
Earth Science; the distance would usually be in either miles or kilometers, and the elevation 
would usually be in feet or meters. Since we will have to use the same units for elevation and 
distance, I will point out to the students that the units will not cancel out and need to be 
included in the answer (i.e. - m/m.). 
 
     After we complete the gradient calculations for A and B I will have the students calculate 
the gradient for A to C, B to C, and A to D. While the students are working I will circulate 
around the room checking results and helping anyone having difficulty. I will work out each 
problem on the overhead when I see that everyone has completed it. 
 
     I will relate gradient to slope and ask the students what they notice about the contour lines 
between A and B, and B and C. I will then have them compare the gradients to the distance 
between the contour lines, and point out that the closer the contour lines the steeper the 
gradient (more change over a small distance) and the further apart the contour lines the more 
gentle the gradient. I will then ask the students if their conclusions holds true for A to C and B 
to D. When the students have responded I will have them use the contour lines to predict 
which gradient is greater, D to E or E to C. The students will calculate the gradients to check 
their predictions. 
 
      
 
 
 
 
 
 3 
 
Mathematics 
 
     The math lesson will begin with a warm-up activity in which students review the concept of 
slope.  We will discuss the similarities between the term gradient used in Earth Science and 
the term slope used in math class.  Then, I will teach students the basics that they need to run 
the Interactive Physics model and make the necessary changes to the parameters.  I will also 
teach students how to make a table of values and a graph using the TI Interactive Software.  
After gathering and displaying the data students will also calculate linear and quadratic 
regressions and decide which one best fits the data generated by the model.    
 
     The work time will proceed with students working in pairs in the computer lab to complete 
the problem below.  As students work, I will monitor their progress by listening to their 
discussions, asking questions, and addressing misconceptions.  At the end of class, students 
will present their findings to the rest of the class and explain how they came to their 
conclusion.   
 
     The model makes the concepts come to life as it would be impossible to actually conduct 
this experiment in the real world.  Furthermore, the software packages make the activity time 
efficient, making it easy to quickly run the experiment, gather the data, and display the results.  
As a result, more time can be spent on discussing the results and their implications.  Students 
are able to make conjectures and within minutes test the validity of their arguments.  The 
model provides a dynamic method allowing students to better understand the link between the 
science and mathematics concepts being studied.   
 
Problem: 
 
1. Using Interactive Physics students will change the horizontal and vertical lengths of an 
incline to model different distances and elevations.  Using the model, students will roll 
an object down the incline.  The model is setup to calculate the velocity of the object 
and automatically pause after 4 seconds. 
 
2. Using Texas Instrument’s Interactive Software, students will construct a table of values 
with four columns: distance, elevation, slope, and velocity.  They will construct a scatter 
plot of the data and calculate linear and quadratic regressions. 
 
3. Students will use the graph and table of values to determine the relationship between 
slope and velocity.  They will discuss which of the regressions best models this 
relationship and present their findings to others.   
 
 
 
 
 
 
 4 
Screenshots of expected student work: 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
Earth Science Rubric: 
 
Category Strong Capable Developing 
Conceptual 
Understanding 
Demonstrates in depth 
understanding of 
concept. 
Demonstrates a general 
understanding of the 
concept. 
Demonstrates limited or 
no understanding of the 
concept. 
Interprets 
Results 
Uses 
inferences/interpretations 
to answer the question 
posed and verify the 
solution. 
Uses limited 
inferences/interpretations 
to answer the question 
posed and verify the 
solution. 
Is unable to interpret 
results or obtain the 
correct solution. 
Communication 
 
Communicates the 
process effectively and 
clearly. 
Communicates a partial 
explanation of the process 
or needs prodding to 
answer questions. 
Communicates a limited 
explanation of the process 
or is unable to 
communicate the process. 
 
 
Mathematics Rubric: 
 
Category 1 2 3 
Use of Interactive 
Physics Software 
 
Did not use IP software 
to simulate the problem. 
IP software was used to 
run the model but the data 
generated was not used 
effectively. 
IP Software was used 
effectively to run the 
model and generate 
data. 
Use of TI Interactive 
Software 
Did not use the Ti 
Interactive software to 
generate a table and 
graph. 
Software was used but the 
data was not displayed 
effectively. 
Software was used 
effectively to generate a 
table and a graph. 
Use of model to 
make  conjectures  
Conclusions are not 
consistent with the model 
and the evidence 
provided is irrelevant. 
Conclusions may be 
inconsistent with the data 
generated by the model 
due to minor errors. 
Able to describe the 
relationship between 
the changes made to the 
initial parameters and 
the resulting 
measurements. 
Accuracy of Results 
 
 
Many flaws in 
calculations, data are not 
organized, or shows no 
understanding of the task. 
Conclusions are 
inaccurate; data could be 
better organized, or shows 
partial understanding of 
the task. 
Calculations are 
accurate, well 
organized, and 
demonstrates full 
understanding of the 
task. 
Application of 
mathematics and 
science concepts and 
understanding of 
their relatedness. 
Does not show any 
understanding of the 
relationship between the 
model and the math or 
science concept being 
studied. 
Partial understanding of 
the concepts and the 
implications of the 
activity. 
Fully understands the 
concepts, is able to 
make generalizations, 
and can apply the 
results of the activity. 
 
 6 
Map used in Earth Science class: 
 
 


Connection Between Gradient and Slope

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Connection Between Gradient and Slope

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