Is My Model Working?

Lesson Content

• Lesson Plan Template:

  General Lesson Plan

• Learning Objectives: What should students know and be able to do as a result of this lesson?

  
  

  Student will be able to:

  • collect quantitative data pairs for the data of their choice.
  • randomly choose two points to save for last part of project.
  • graph data pairs as a scatter plot with an appropriate domain and range.
  • determine the line of best fit for their data.
  • interpret the slope and y-intercept in context of their data.
  • draw conclusions about the consistency of their line of best fit (or prediction equation) based on examining residuals.

• Prior Knowledge: What prior knowledge should students have for this lesson?

  
  

  Prior knowledge should include:

  • calculating slope when given two points.
  • writing the equation of a line given two points.
  • determining a line of best fit for a scatter plot.
  • graphing a scatter plot and using appropriate scales for the axes.
  • interpreting the slope and y-intercept in the context of a real world problem.

• Guiding Questions: What are the guiding questions for this lesson?

  
  

  Students should be excited about collecting the data of their choice. It should be emphasized that there are no wrong data pairs to choose as long as the data is quantitative.

  What types of data are represented by a linear relationship?

  What roll do algebraic models play in statistics?

• Teaching Phase: How will the teacher present the concept or skill to students?

  
  

  This lesson is a project that will be completed over several days. Teacher may choose for students to complete most of it in class or to assign parts of it to be completed outside of class. Teacher should refer to the Is My Model Working project pages that are included as an attachment. Also attached is a power point Is My Model Working PPT that can be used by the teacher in the classroom each day as new parts of the project are started.

  Day 0. 10 minute Formative Assessment. Teacher will give students a scatter plot and ask them to find the equation for the line of best fit and give an explanation of the slope and y-intercept in context of the data.

  Day 1. 15 minute project introduction. Teacher will introduce the project as an opportunity for students to study data around them. Students will be asked to choose quantitative data pair ideas from a list (see project handout and examples below) or to use their own idea for data pairs. (Note: The data collection could be done in class but only if data pairs are restricted to data that students can get from classmates, or the Internet, if available in classroom.)

  Examples from project handout:

  • Shoe size and height in inches for 20 friends (must be same sex because of differences in shoe sizes)
  • Weight bench-pressed and height in inches of 20 friends
  • Miles from home and time getting to school for 20 friends

  Homework. Data collection: Students need to choose their data idea and then collect 20 data pairs from chosen sources. Teacher should make suggestions but let students be responsible for finding their data. Students should enter their collected data in the table in the project handout.

  Day 2. 10 minutes to check on student data. Students now need to randomly select two of their points to be saved for the last part of project. They will be used to evaluate the predication equation.

  To randomly select two points of the 20, teacher could have a container with the numbers 1-20 on slips of paper. This container can be passed around the room with each student drawing and recording two numbers. These will be the points that will be saved until last section of project. Students need to remember to return the numbers they chose to the container and pass it to next student.

  If teacher prefers, a random number generator on a calculator or computer could be used.

  Teachers, should remind students how to set up their scatter plot graph. Emphasize that this is a first quadrant graph. Why? (Because domain and range of data are positive numbers.) Students should be advised to set up their units on the x-axis and the y-axis so that the majority of the graph is used. Why? (Because spreading out the points will make finding the line of best fit easier and allow the students to see the trend of the data.) Students should be reminded how to choose and/or show unit breaks in graphs.

  Teacher can use the attached Is My Model Working PPT with prepared sample or take sample data from one of the students to review with the class how to choose an appropriate scale for each axis. Teacher should emphasize that students are to graph only 18 points. Teacher could review again how to find a good line of best fit.

  Classwork or Homework: Students should complete their scatter plots using their 18 points. The points should be plotted on the supplied graph in the project pages. Students should then scrutinize their graphs and draw in a line of best fit. (10-15 minutes)

  Day 3. 15-20 minutes for teacher to check on the students' individual scatter plots with line of best fit. Then teacher should model finding the equation of the line of best fit using two points on the line. Is My Model Working PPT example can be used or some data from a student in class. Mention that the project pages have an area for this work to be shown.

  Teacher should continue discussion giving an interpretation of the slope and y-intercept for the students' best fit line. The teacher should remind the students to use appropriate units from their data for the slope and y-intercept. Also discuss whether or not the y-intercept is always meaningful. This should lead to a discussion of reasonable domain.

  Classwork or Homework: Students should find an equation for their line of best fit and write full sentence interpretations of their slope and y-intercept. Student work should be recorded on the project pages. (15-20 minutes)

  Day 4. 15-20 minutes. Teacher should check to see if the students have been successful in finding the equation of the line of best fit and in evaluating their slope and y-intercept. Since it is time consuming to check every step, suggest that seat partners check each other.

  After this check, teacher will help students to determine whether or not their line of best fit is a good prediction equation. This will be done by looking at the residuals, i.e., the differences between the corresponding collected or observed y values and those predicted by the line of best fit. The term residual may be a new word.

  Definition: Residual = Observed Value - Predicted Value. Note that each data point has a residual and are sometimes represented by a residual plot.

  Teacher should explain that this is why the two random data pairs were saved. Teacher could model this with a student's sample data and their line of best fit, or use prepared example in Is My Model Working PPT. Teacher should have the students plug in the x values for the extra two points into the prediction equation, then subtract the results from the collected (or observed) y values from these points. These differences are the residuals. The project page is set up to help students do this calculation.

  Teacher needs to help students understand how the residuals provides information regarding the consistency of their data. If the differences are small in context of the data then the line of best fit appears to be a good prediction equation or model. If the residuals are large in comparison to the context of the data then the line of best fit is not a good prediction equation. If one residual is small and the other large then no conclusion can be made at this time other than that more data should be collected to make a conclusion. (The size of the residuals can vary in context of the data, e.g., if the context is students' weight, 5-8 pounds would be a small difference but over 15 pounds would be a large difference.)

  Classwork or Homework: Students should find their residuals and write an evaluation or conclusion determining whether or not their line of best fit is a good predication equation or model. This should be entered into the project paper.

  Students now need to check to make sure that all of their project parts are complete and neatly presented so that it can be turned in.

  Day 5. 15-20 minutes. Students are to turn in their completed projects. Teacher may choose a few students (or students can volunteer) to present details of their project and what they learned. For closure, students as a group should come up with characteristics of a good model.

• Guided Practice: What activities or exercises will the students complete with teacher guidance?

  
  

  As noted in the teaching phase, teacher can choose to have students complete parts of project in class with teacher guidance.

• Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?

  
  

  As noted in teaching phase, teacher can decide that students work on parts of the project independently in class or complete for homework.

• Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?

  
  

  Students have individually organized the knowledge needed from the project in their written conclusion regarding the consistency of their model or prediction equation. When projects are collected, teacher could select students, as time permits, to present their findings.

  Teacher should also question students about their understanding of behavior in real world data and have the class discuss how models can be useful in the real world.

• Summative Assessment

  
  

  The summative assessment will be the successful completion of the project with emphasis on valid conclusions made by the student concerning the linearity of their chosen data.

  Here is suggested rubric for the project:

  GRADING RUBRIC:

  Part 1: T-chart with 20 well-labeled data pairs & sources up to 20 points____

              (note: due on earlier date)

  Part 2: Well-labeled correct graph with 18 points & best fit line

               drawn in.                                                                         up to 20 points____

  Part 3: Correct work shown to find slope, y-int. & equation of line     up to 20 points____

  Part 4: Clear interpretations of slope & y-intercept in sentence form   up to 20 points____

   

  Part 5:  Correct evaluation shown for extra two points and residual with clear conclusion

            and explanation about prediction equation/model.                             up to 20 points____

   

  Extra Credit:  Neat work.                                                                  up to 5 points ____

   

                                                                                                 

                                                                                                    Total  (up to 105)        _______

   

• Formative Assessment

  
  

  Day 0. At the beginning of this lesson it is recommended that students should be quizzed formally or informally on their knowledge of finding an equation for the line of best fit from a scatter plot.

  Example: For the following scatter plot find an equation for a line of best fit. Also, give an explanation of slope and y-intercept in context of data. This example with questions and answers is included in Is My Model Working PPT that is attached.

  

  Other informal formative assessments will completed by students throughout the 5-day project. These are discussed in the Teaching Phase.

• Feedback to Students

  
  

  During this project activity, feedback will be given to students on a daily basis as they proceed through parts of the project. Teacher will discuss progress with the class and individually check on:

  Part 1: Data collection. Teacher will check to make sure that data is appropriate, e.g. expressed height in inches and don't mix boys and girls shoe sizes.

  Part 2: Drawing of scatter plot. Have students labeled the graphs correctly? Did they choose appropriate axes' scales? Did they find a line of best fit using trend of data? Did they randomly take out two points to save for later?

  Part 3: Calculation of line of best fit. Do students remember how to find slope and y-intercept? Did they find the correct prediction line or algebraic model? Ask students if the line equation that they found makes sense when looking at their graph, e.g. Is the slope positive? Will the y-intercept hit about where they think?

  Part 4: Slope and y-intercept interpretation. Did students understand how to interpret these parameters? Can they put it in a sentence?

  Part 5: Evaluation of the line of best fit to be used as a prediction equation or model. Did students find expected values and use with actual values to find residuals? Do they understand how to use residuals to make a conclusion about their prediction equation? Did they show appropriate work?

  This check should take 10-15 minutes daily during the week of the project.