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Estimating With Fire
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Abstract

In this lesson, students will investigate the various methods and functions of estimation by modeling the spread of a forest fire.

Objectives

Upon completion of this lesson, students will:

  • understand the relative merits and uses of estimating using fractions, percents, and numbers
  • understand how the uses of estimation depend on the context
  • gain experience applying different methods by which estimation can be conducted more quickly and more accurately

Standards Addressed:

Student Prerequisites

  • Arithmetic: Students must be able to:
    • convert between percents, fractions, and decimals
    • calculate and understand probability through percents, fractions, and decimals
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

algorithmStep-by-step procedure by which an operation can be carried out
chaosChaos is the breakdown of predictability, or a state of disorder
decimalShort for the term "decimal fraction", a decimal is another way to represent fractional numbers. The decimal uses place value to express the value of a number as opposed to a fraction that uses a numerator and denominator.
fractionA rational number of the form a/b where a is called the numerator and b is called the denominator
percentA ratio that compares a number to one hundred. The symbol for percent is %
probabilityThe measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability

Lesson Outline

  1. Focus and Review

    Remind students of what they have learned in the past that is relevant to the situation at hand. Ask the following questions:

    • What does it mean to estimate the answer to a problem?
    • How does this differ from finding an exact answer?
    • Raise your hand if your birthday is between January 1 and June 31.
      • How many people in this school do you think have birthdays in the first half of the year?
      • If you were estimating that, what are some different ways you could phrase your answer? Could you use a number? A fraction? A percent?

  2. Objective

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today we are going to investigate the relative merits of using fractions, percents, and decimals to estimate trees burned in a fire. Then, we will discuss how different algorithms can be used to make estimating easier, faster, and more accurate in certain situations.
    • We will be using a computer to simulate burning down a forest and check our estimations, but please do not turn your computers on until I tell you to do so.

  3. Teacher Input

    Discuss the concept and utility of estimation. Ask the following questions:

    • What are some situations in which estimation would be more useful than actually counting out an exact number?
    • There are three main ways to express estimated results - numbers, fractions, and percents
      • Can anyone describe how to estimate the number of objects with a certain property if the objects are difficult to count?
      • What about the fraction of some objects with a certain property?
      • What about the percent of some objects with a certain property?
      • Can we convert between these three measures? How?

  4. Guided Practice

    Show the students the Fire Assessment applet. Set one of the trees on fire and watch the fire spread. Pose the following questions to the students:

    • About how many trees do you think burned?
    • About what proportion of the trees do you think burned, as a fraction?
    • About what proportion of the trees do you think burned, as a percent?
    • Which type of estimation was easiest - number, fraction, or percent?
    • How close do you think your estimation was?

    Put the class's estimate into the "Guess the Burn" box after first specifying whether you are entering numbers, percents, or fractions. Submit to check how close the class actually was.

    Demonstrate the different controls of the applet to students, including the following:

    • How to change the percentage chance that a fire will spread
    • How to change the forest size
    • How to change between estimating percents, fractions, and numbers

  5. Independent Data-Collection

    Have students work in groups of 2-4 to fill out the worksheet. Assign each group a certain burn probability and forest size, then have them fill out the worksheet, estimating fractions, percents, and numbers of trees burned. As they work, instruct students to think about which type of estimation was easier/more accurate and why.

    When groups finish with the worksheet, have them write responses to the exploration questions to reflect on their work.

  6. Discussion

    Bring the class back together and discuss their responses to each of the exploration questions. Have students write their average error in estimation for percents, fractions, and numbers on the board. Based on those results, come to a conclusion as a class as to which type of estimation is the most accurate.

    Discuss with students the different methods of estimation and reasons why each may or may not have worked well in this exercise.

    Discuss other contexts for estimation and how the requirements can be different. Ask each group to come up with a situation in which their estimation algorithm would be most appropriate and discuss the various situations.

Alternate Outline

This lesson can be rearranged in the following ways:

  • Have students first attempt to estimate burn percentage without algorithms, and then introduce the Divide-and-Conquer, Block Sampling, and Subset-Whole methods of estimation.
  • Give students examples of various real-life situations (such as election results, road lengths, grains of sand, etc) where estimation might be used, and ask students to identify which type of estimation and what degree of accuracy is needed in each situation.

Suggested Follow-Up

To follow up on the development of estimation skills, use the Estimation lesson and activity.

To shift focus towards various conceptions of probability, use the Conditional Probability of Simultaneous Events or Unexpected Answers lessons.

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