Testing Bridge Lengths

1
Testing Bridge Lengths

• The Gadsden Group

2
Goals and Objectives

• Collect and express data in the form of tables
  and graphs
  
• Look for patterns to make predictions from tables
  and graphs
  
• Distinguish between linear and non-linear
  relationships from tables and graphs

3
Standards and Benchmarks

• Standard 2 Algebra
• Definition Students will understand algebraic
  concepts and applications.
  
• Benchmark 2-D
• Analyze changes in various contexts
• 1. Use graphs, tables, and algebraic
  representations to make predictions and solve
  problems that involve change.
  
• 2. Estimate, find, and justify solutions to
  problems that involve change using tables,
  graphs, and algebraic expressions.
  
• 3. Use appropriate problem-solving strategies
  (e.g., drawing a picture, looking for a pattern,
  systematic guessing and checking, acting it out,
  making a table or graph, working a simpler
  problem, writing an algebraic expression or
  working backward) to solve problems that involve
  change.
• 5. Analyze problems that involve change by
  identifying relationships, distinguishing
  relevant from irrelevant information, identifying
  missing information, sequencing, and observing
  patterns.
• 7. Recognize the same general pattern of change
  presented in different representations.

4
Standards and Benchmarks contd

• Standard 5 Data Analysis and Probability
• Definition Students will understand how to
  formulate questions, analyze data, and determine
  probabilities.
  
• Benchmark 5-A
• 1. Represent two numerical variables on a plot,
  describe how the data points are distributed and
  identify relationships that exist between the two
  variables.
• 3. Organize, analyze, and display appropriate
  quantitative and qualitative data to address
  specific questions including plots charts and
  tables.
• 5. Simulate an event selecting and using
  different models.
  
• Benchmark 5-B
• 3. Analyze data to make decisions and to develop
  convincing arguments from data displayed in a
  variety of formats that include graphs scatter
  plots charts and tables
• 4. Interpret and analyze data from graphical
  representations and draw simple conclusions
  (e.g., line of best fit).
  
• 5. Evaluate and defend the reasonableness of
  conclusions drawn from data analysis.
  
• 7. Identify simple graphic misrepresentations
  and distortions of sets of data (e.g., unequal
  interval sizes, omission of parts of axis range,
  scaling).

5
Standards and Benchmarks contd

• Benchmark 5-C
• Develop and evaluate inferences and predictions
  that are based on data
  
• 2. Describe how reader bias, measurement errors,
  and display distortion can affect the
  interpretation of data, predictions, and
  inferences based on data.
• 3. Conduct simple experiments and/or
  simulations, record results in charts, tables, or
  graphs, and use the results to draw conclusions
  and make predictions.
• 4. Compare expected results with experimental
  results and information used in predictions and
  inferences.
  

6
Lesson Non-Linear Models

• In this investigation, students will encounter
  a different kind of relationship an inverse
  relationship. In an inverse relationship, as one
  variable decreases the other increases, but not
  in a linear fashion - that is, not by constant
  decreases. There is an underlying constant in
  inverse relationships when the two variables
  are multiplied, they yield a constant product.
  Students will see that the pattern of change in
  the data is similar to that in the other inverse
  relationships they will investigate. (Lappan, et
  al, Thinking With Mathematical Models, p. 36a).

7
Lesson Contd

• Students will explore non-linear relationships by
  testing the lengths of bridges and their breaking
  weight.
  
• Launch
• What do you expect to happen in this experiment?
• We are using equipment similar to what we have
  used before. What are the variables this time?
  (Length and breaking weight)
  
• What will the data look like? What shape do you
  think the graph will be?

8
Lesson contd

• Explore
• Have students conduct the experiment and discuss
  the questions within their group.
  
• Ask each student to make a table/graph and write
  his or her own answers.
  
• Each group will compare their data and come to a
  consensus in order to create a group poster of
  their table and graph.

9
Lesson contd

• Summarize
• After students have displayed and presented their
  group work, then ask
  
• Are there similarities in the results of the
  various groups? Are there differences? What
  might have caused those differences?
  
• As bridge length increases, what happens to the
  number of pennies that can be supported? (It
  decreases)
  
• As bridge length decreases, what happens to the
  number of pennies that can be supported? (It
  increases)
  
• Focus the classs attention on this inverse
  relationship.
  
• What shape or pattern do you see in your graph?
  Is it linear? (no, it is definitely a curve)
  
• How might you be able to tell from your table
  that the graph model might be a curve? (The
  difference between two consecutive breaking
  weights decreases less and less as the length
  increases it is not a constant change)