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Lesson Plan: Continuous vs Discrete Data

Standards 

  • CA Investigation and Experimentation (multiple grade levels): Select and use appropriate tools and technology to perform tests, collect Data, analyze relationships, and display Data

Overview

When working with Data, it is important to know whether the Data is continuous or discrete, as this will determine what calculations can be applied and what Information can be extracted from it. This lesson helps students to distinguish between continuous and discrete data and explain why that matters.

Prerequisites: None

Materials:

  • Projector
  • Computers connected to Internet

Suggested Time Frame:

Terminology

  • Aggregate: Grouping Information together in a way that makes it easy to interpret.
  • Discrete: Data that can be known and counted exactly (# of pens in a box).
  • Continuous: Data from measurements that cannot be known exactly (temperature of the room).
  • Range: The upper and lower bound on a set of Data. Necessary to talk about continuous Data.
  • Precision: How close Data is to the correct value (e.g. 1.001 is closer than 1.01 to 1).

Activity: Distinguish Between Continuous and Discrete Data

Teacher Notes:

Slides 1 and 2  show the difference between discrete and continuous Data. Slide 3 clarifies why continuous Data can never be exact. Between any two numbers on the real number line, there is always an infinite amount of numbers. Therefore, continuous Data can only be shown as a range and not as exact points.

Slides 4-7 assess, review, and clarify any confusion between continuous and discrete Data.

Activity: Collecting Discrete and Continuous Data

  1. Share the continuous and discrete survey and have your students fill it out or have them write their answers on the board.
  2. Ask the students to vote on each question, whether it is discrete or continuous.

Teacher Notes:

  • Age: Continuous (can be broken into months, days, hours, seconds, milliseconds, etc)
  • Hair Color: Discrete (as long as everyone agreed upon the colors)
  • Height: Continuous
  • Favorite Color: Discrete
  • Do you own a pet? Discrete

Student Questions:

  1. What would happen if instead of choosing the color from a list you could choose it from a color picker, would it still be discrete? A: Yes, each of those colors are assigned a specific value.
  2. Based on your class’ data, how likely is it that a new student:
  1. Likes the color blue?
  2. Is older than the class average?
  3. Owns a pet?



Activity: Modeling Continuous Functions

Subactivity: Speed of Light

Teacher Note:

In this activity, students examine historical measurements of the Speed of Light. We refer to the speed of light as a constant at 300,000 km/sec, or more precisely, 299,792 km/sec. How did we discover that number? Like pi, the Speed of Light has been calculated with increasing precision, and over time, that number seems to converge towards 299,792 km/sec.

By working with this data, students will gain the skill of aggregating large amounts of Data while using it to confirm a fundamental constant of the universe.

  1. Share the Speed of Light Fusion Table with your students.

Teacher Notes:

A Fusion Table is like a spreadsheet for large amounts of Data. This table is based on experiments conducted throughout the years.

  1. Have the students spend one minute looking over the Data and write down their observations. Afterwards, have the students look away from their computers and share what they noticed.

Student Questions:

  1. What patterns do you see? What do the numbers have in common and what do they differ on?

A: Numbers are relatively large, numbers are close together, some have decimals while others do not, numbers get progressively more precise as time goes on.

CT: Students use technology to see patterns more easily within data.

  1. Would this data be continuous or discrete? A: Continuous. If they say discrete it may be because these are numbers written in a table.
  1. Remind them that these were experiments conducted by scientists over time to measure the speed of light. Velocity is distance divided by time, which can never be measured exactly, therefore the Data is continuous.
  2. Remind the students that continuous Data means that all Data is possible for a certain range of numbers (e.g. one’s height is 6 ft ± margin of error).

  1. Have the students click on options and then Aggregate.
  2. As in the pictures below, select average, maximum, and minimum for each year and Apply.

Teacher Notes:

The students should see the calculated values for each year. They may not understand the significance of what they have done, so remind them that, in less than a second the computer has sifted through 600 points of Data to make these calculations. Imagine having to do these calculations by hand! In the past, scientists and mathematicians had no choice but to calculate large amounts of Data by hand, and were it not for their patience, we would be centuries behind in technology and discovery.

  1. To make it even easier to see the Information, have students click on Visualize in the top left, and from the drop down menu, click Scatter.
  2. Set the X Axis to Count. For the Y Axis hold down the Shift  key while you select each year’s Maximum, Minimum, and Average (do not select count at the bottom).

Student Question:

  • What do you notice about this graph? What does it tell us about the value of c (speed of light)? A: As the year the Data was measured (and therefore the precision of those measurements) increases, it converges closer and closer to the accepted constant value for c.

CT: Students are able to generalize their understanding of the data to make predictions.

Subactivity: The Uncanny Valley and Drawing a Circle

Teacher Notes:

Computers have improved dramatically over the last 20 years and every industry has seen significant increase in what they are able to do. One sector that has found an upper bound to what they can do is the entertainment industry. Computer graphic artists and designers are able to make better looking images and animations because the computers are able to handle more complex geometry at a high speed. However, an interesting thing occurred in 2001 when the popular game Final Fantasy was released as a movie.

While it was not the first movie to be made with 100% computer graphics, it was by many accounts the most realistic, and this lead to viewers feeling “awkward”. This has been referred to as the Uncanny Valley (NPR Article). It describes a limit of similarity people are able to tolerate in non-human analogs (e.g. video, robots, etc).

Computer graphics is all about geometry, memory, and speed. A virtual object is made up of shapes tessellated and oriented until they follow the contours of the real model. The more sides the object has, the smoother the object can look.

Pictures from Wikipedia

A computer makes a 2D curve or a circle by taking two points and drawing a line between them. If the distance between the points is small enough, then it looks curved to real to the viewer. When creating objects on a computer, we need to decide how realistic we need it to be. With movies, realism is key, but a game where speed is most important, a overly detailed model can use up all of the memory.

For this activity, students can use Python, Geogebra, or a piece of paper.

Python Instructions:

  1. Have students predict what the maximum turning angle can be
  2. Open the Python Interpreter and type:

from turtle import *         #For simple drawings in Python.

  1. You have 2 functions available that you can use to create your circle.

                 forward(x)        #Moves the turtle forward x number of pixels

                right(ϴ)         #Turns ϴ degrees right. There is also left(ϴ)

  1. See what is the largest you can make x and ϴ to create a seemingly smooth circle.
  2. It might be helpful to use the functions:

        undo()                #Removes the last instruction

        reset()        #Erases everything in and goes back to home

Here are two examples:

from turtle import *

forward(100)

right(45)

forward(100)

right(45)

forward(100)

        

from turtle import *

forward(50)

right(30)

forward(50)

right(30)

forward(50)

right(30)

forward(50)

        Geogebra Instructions:

  • Use the Segment Between Two Points tool to determine the maximum distance and angle between the points where it looks like a curve.


        Paper and Pencil Instructions:

  • Using the ruler, create points that are a fixed distance from each other and connect them with a straight line. Experiment with the distance and angle between the points until it looks like a curve.

Teacher Notes:

The uncanny valley activity is perfect for laying the conceptual groundwork for calculus and limits. Students appreciate hearing that the games they play and movies they watch have math behind them.

There isn’t a correct answer in this case as far as the distance and angle between the points. To one person it may look like a curve while to another the illusion fails. This is why people who work with this type of graphics are considered by their peers to be artists.

Student Questions:

  1. Before starting this activity, how many points do you predict you will need to draw? At what distance and angle?
  2. After completing this activity compare your results with your predictions.
  3. Compare your drawing with 3 other students, how do yours compare (not in terms of quality) but do they look like a curve as well?
  4. Why do computer graphic artists use polygons like triangles and hexagons to draw curved surfaces? A: It creates the appearance of 3D Curves while using less memory.
  5. How would you tell the computer to draw the perfect circle? Could it be done? If you were to tell the computer to draw a square, then a pentagon (5 sides), hexagon (6 sides), septagon (7 sides), and so on would there be a pattern to your instructions? What would it be? A: Although this is technically impossible, it would require an infinite number of sides and turns.

CT: Students use technology to decompose a circle into a polygon with an infinite number of sides.

Assessments

  • At the end of this unit, students will be able to differentiate between Continuous and Discrete Data. A multiple choice test would be the easiest way to assess understanding. Provide examples of Data, charts, or scenarios with a options of Continuous or Discrete. In order to minimize misconceptions moving forward, each answer could be accompanied by a sentence or two defending their answers. Students could discuss their responses after the test and ask remaining questions.
  • For the Python/Geogebra/Paper activity, students should submit at least two drawings/outputs with different distances and a short explanation of how they determined the optimum distance.

Other Resources

  • Two interesting stories about Continuous Data with very different outcomes are:


More lessons and examples can be found at Google’s Exploring Computational Thinking website.

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