© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 1 Problem Solving and Critical Thinking

© 2010 Pearson Prentice Hall. All rights reserved Estimation, Graphs and Mathematical Models

© 2010 Pearson Prentice Hall. All rights reserved. 3 Objectives 1.Use estimation techniques to arrive at an approximate answer to a problem. 2.Apply estimation techniques to information given by graphs. 3.Develop mathematical models that estimate relationships between variables.

© 2010 Pearson Prentice Hall. All rights reserved. Estimation The process of arriving at an approximate answer to a question. Methods of Estimation Rounding Numbers Using Graphs Using Mathematical Models 4

© 2010 Pearson Prentice Hall. All rights reserved. Rounding Whole Numbers 5 The symbol ≈ means “is approximately equal to”

© 2010 Pearson Prentice Hall. All rights reserved. 6 Example 1: Rounding A Whole Number Round the World population 6,751,593,103 –To the nearest million: –To the nearest thousand:

© 2010 Pearson Prentice Hall. All rights reserved. 7 Example 1: Rounding A Whole Number Round the World population 7,142,003,937 (as of 1/21/14 10:07 a.m.) –To the nearest million: –To the nearest thousand: US Population: 318,892,103 (2013) Hawaii Population: 1,404,054 (2013)

© 2010 Pearson Prentice Hall. All rights reserved. 8 Example 2: Rounding The Decimal Part of a Number The first seven digits of π are: Round π: –To the nearest hundredth: –To the nearest thousand:

© 2010 Pearson Prentice Hall. All rights reserved. Archimedes (250 BC) pi ≈ 22/7 ≈ π c d π = c / d

© 2010 Pearson Prentice Hall. All rights reserved. Calculate perimeters of in circumscribed and inscribed polygons Archimedes (250 BC) 223/71 < pi < 22/7 10 π

© 2010 Pearson Prentice Hall. All rights reserved. 11 Example 3: Estimation by Rounding You purchased bread for $2.59, detergent for $2.17, a sandwich for $3.65, an apple for $0.47 and coffee for $5.79. The total bill was $ Is this amount reasonable? To check, round the cost of each item to the nearest dollar. This bill of $18.67 seems high compared to the $15.00 estimate. Actual Cost: $14.67

© 2010 Pearson Prentice Hall. All rights reserved. 12 Example 5: Applying Estimation Techniques to a Circle Graph Use a circle to represent the whole and divide it into sectors proportional to their part of the whole. According to the U.S. Census Bureau, in 2008, there were 220,948,915 Americans 20 years and older. Assuming the circle graph is representative of this age group, a. Determine a calculation that shows the number of Americans 20 years and older who define old age by a decline in physical ability. b. Use rounding to find a reasonable estimate.

© 2010 Pearson Prentice Hall. All rights reserved. 13 Example 5 continued b. Estimate the Answer Approximately 88 million Americans 20 years and older define old age by a decline in physical ability. a. Compute the exact answer

© 2010 Pearson Prentice Hall. All rights reserved. 14 Estimation with Graphs: Bar Graphs Bar graphs use a vertical or horizontal bar to represent each item. The length of the bar determines the amount. We can use bar graphs to make predictions.

© 2010 Pearson Prentice Hall. All rights reserved. 15 Example 6: Applying Estimation and Inductive Reasoning to Data in a Bar Graph a. Estimate a man’s increased life expectancy, rounded to the nearest hundredth of a year, for each future birth year. For each subsequent birth year, a man’s life expectancy is increasing by approximately 0.19 year.

© 2010 Pearson Prentice Hall. All rights reserved. 16 b. Estimate the life expectancy of a man born in An American man born in 2020 will have a life expectancy of approximately 78 years. Example 6: Applying Estimation and Inductive Reasoning to Data in a Bar Graph continued

© 2010 Pearson Prentice Hall. All rights reserved. 17 Estimation with Graphs: Line Graphs Line graphs illustrate trends over time Horizontal axis represents time Vertical axis represents the age

© 2010 Pearson Prentice Hall. All rights reserved. 18 Example 7: Using a Line Graph Find an estimate for the percentage of seniors who used marijuana in In 1990, approximately 14% of seniors used marijuana.

© 2010 Pearson Prentice Hall. All rights reserved. 19 Example 7: Using a Line Graph continued In which five-year period did the percentage of seniors who used marijuana decrease at the greatest rate? The percentage of seniors using marijuana decreased at the greatest rate in the five-year period from 1985 through 1990.

© 2010 Pearson Prentice Hall. All rights reserved. 20 Estimation with Mathematical Models The process of finding formulas to describe real- world phenomena.

© 2010 Pearson Prentice Hall. All rights reserved. 21 Example 8: Modeling the Cost of Attending a Public College a. Estimate the yearly increase in tuition and fees. The average yearly increase is about $353.

© 2010 Pearson Prentice Hall. All rights reserved. 22 Example 8: continued b. Obtain a mathematical model that estimates the average cost of tuition and fees, for the school year ending x years after 2000.

© 2010 Pearson Prentice Hall. All rights reserved. 23 Example 8 continued c. Use the mathematical model to project the average cost of tuition and fees for the school year ending in The model projects that the average cost of tuition and fees for the school year ending in 2014 will be $8304.