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:

Grade 3

Estimation and Computation

The student determines reasonable answers to real-life situations, paper/pencil computations, or calculator results.

Numeration

The student demonstrates conceptual understanding of whole numbers up to one thousand.

Grade 4

Estimation and Computation

The student determines reasonable answers to real-life situations, paper/pencil computations, or calculator results.

Numeration

The student demonstrates conceptual understanding of whole numbers to ten thousands.

Grade 5

Estimation and Computation

The student determines reasonable answers to real-life situations, paper/pencil computations, or calculator results.

Numeration

The student demonstrates conceptual understanding of whole numbers to millions.

Grade 6

Estimation and Computation

The student determines reasonable answers to real-life situations, paper/pencil computations, or calculator results.

Grade 7

Estimation and Computation

The student solves problems (including real-world situations) using estimation.

Grade 8

Estimation and Computation

The student solves problems (including real-world situations) using estimation.

Number and Quantity

Quantities

Reason quantitatively and use units to solve problems.

Grades 3-5

Measurement

Apply appropriate techniques, tools, and formulas to determine measurements

Grades 6-8

Measurement

Apply appropriate techniques, tools, and formulas to determine measurements

Grade 6

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.

3rd Grade

Measurement

The student will demonstrate through the mathematical processes an understanding of length, time, weight, and liquid volume measurements; the relationships between systems of measure; accurate, efficient, and generalizable methods of determining the perimeters of polygons; and the values and combinations of coins required to make change.

3rd Grade

Measurement

Content Standard 4.0 The student will become familiar with the units and processes of measurement in order to use a variety of tools, techniques, and formulas to determine and to estimate measurements in mathematical and real-world problems.

4th Grade

Measurement

The student will become familiar with the units and processes of measurement in order to use a variety of tools, techniques, and formulas to determine and to estimate measurements in mathematical and real-world problems.

5th Grade

Measurement

The student will become familiar with the units and processes of measurement in order to use a variety of tools, techniques, and formulas to determine and to estimate measurements in mathematical and real-world problems.

4th Grade

Computation and Estimation

4.5 The student will estimate whole-number sums and differences and describe the method of
estimation. Students will refine estimates, using terms such as closer to, between, and a
little more than.

6th Grade

Measurement

6.10 The student will estimate and then determine length, weight/mass, area, and liquid volume/capacity, using standard and nonstandard units of measure.

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

Step-by-step procedure by which an operation can be carried out

chaos

Chaos is the breakdown of predictability, or a state of disorder

decimal

Short 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.

fraction

A rational number of the form a/b where a is called the numerator and b is called the denominator

percent

A ratio that compares a number to one hundred. The symbol for percent is %

probability

The 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

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?

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.

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?

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

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.

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.