Month Rabbits # Pairs


 Opal Spencer
 3 years ago
 Views:
Transcription
1 1. Fibonacci s Rabbit Problem. Fibonacci rabbits come in pairs. Once a pair is two months old, it bears another pair and from then on bears one pair every month. Starting with a newborn pair at the beginning of a year, how many pairs of rabbits will there be at the end of the year? I start the problem in the pictures below. A white pair if rabbits is one that is not yet mature enough to produce another pair. Pink pairs reproduce. It may be hard to know who the parents are after a while, but for every pink pair in a given row, there will be an additional white pair in the row below. Page from the Liber Abaci where the rabbit problem appears Month Rabbits # Pairs After a year there will be rabbits.
2 2 Solution. One way of doing it is to see the pattern, or to just go on with the rabbit pictures until month 12 is reached. We then get the following pattern: Month J F M A M J J A S O N D # rabbit pairs But suppose we want to go on for longer than a year, can we be sure the pattern stays the same? Here is a more precise justification. It may be a bit harder to follow, but I hope not incredibly hard. Let us look at some consecutive months and following the mathematical custom of calling things by letters, suppose that one month is called n, the next one n + 1. Suppose that in month n we have x pink rabbit pairs and y white ones. We write this out as p = x, w = y. The total number of rabbit pairs is then fn = x + y.(fn for Fibonacci at time n). Now every pink rabbit pair will produce a new pair that is white. The white pairs become pink. This means that in month n + 1 our equations are p = x + y (all pink stay pink, whites become pink) and w = x (all pink produce one white). So now we have for the total number of rabbit pairs f(n + 1) = x + y + x = 2x + y. This still doesn t tell us that much, so let s go to the next month, month n + 2. All pink stay pink, all white become pink, our equation is now p = x + y + x = 2x + y; all pink produce one white, so w = x + y, and f(n + 2) = 2x + y + x + y = 3x + 2y. Well, tis is still a bit obscure, but we might notice that f(n + 2) = 3x + 2y = (2x + y) + (x + y) = f(n + 1) + fn. The pattern is, in fact and forever: The number of rabbit pairs in any given month past the second one equals the sum of the number of pairs in the the two preceding months. That is, the pattern is 1, 1, = 2, = 3,, = 5,, = 8, = 13, and so forth The sequence of numbers so created is known as the Fibonacci sequence. 2. A tiling problem. Suppose we have rectangular tiles; they are twice as long as they are wide.to be precise, each tile is 10 inches long and 5 inches wide. Mr. Snuckelbakker wants to use these tiles to tile a 10 inch wide rectangular area in a room of his house and he wonders in how many different ways he can lay the tiles, depending on how much floor he wants to cover. If all he wants to do is to cover 5 inches of floor, there is only one way of doing it, he uses a single tile: If he wants to cover 10 inches of floor, he has two choices:
3 3 If he wants to cover 15 inches of floor, he has three choices: However, he wants to cover a strip that is 10 feet = 24 5 long. In how many ways can he do it? Solution. The Fibonacci sequence appears again, except that the first term is missing. The sequence is 1, 2, 3, 5,.... So the answer is the 25th number in the series, namely f 25 = 75, 025. Here is a justification: Say Mr. S has laid out tiles covering a strip at least 15 long. The tiling either ended with a single vertical tile (as in the first and third picture of how to tile a 15 inch strip) or with two horizontal tiles (as in the middle picture of how to tile a 15 inch strip). In the first case, removing the last tile, tiles 5 inches less, in the second case, removing the horizontal tiles, tiles 10 inches less. From this one can see that the number of different ways to tile a strip of n times 5 inches is the sum of ways one can tile a strip of n 1 times 5 inches, plus the number of ways one can tile a strip of n 2 inches. Fibonacci again! 3. A problem posed to Fibonacci by a magister from near Constantinople. This is one of a whole series of problems involving two or more men, giving and taking money. You can replace the word denari by dollars. Also it is proposed that one man takes 7 denari from the other, and he will have five times the second man. And the second man takes 5 denari from the first, and he will have seven times the denari of the first. How much money does each have? Modern version: If A were to take $7 from B, then A would have five times what B has. If B were to take $5 from A, then B would have seven times what A has. How much money do A, B have? The correct answer involves fractions! Solution. This problem involves equations in addition to fractions. Let us call a the amount of money A has, b the amount B has. The problem states: b + 5 = 7(a 5). a + 7 = 5(b 7) The first equation can be worked on a bit to get first a + 7 = 5b 35, we can then subtract 7 from both sides to get a = 5b 42. We go with this to the second equation: b + 5 = 7a 35 = 7(5b 42) 35 = 35b = 35b 329.
4 4 We can turn this equation around to get 35b 329 = b + 5; subtracting b from both sides gives 34b 329 = 5, adding 329 to both sides gives 34b = 334. From this: b = = 167 Returning with this value of b to the equation a = 5b 42 we get a = = = 835 The answer is that A has 121/, B has 167/ denarii. 42 = = Fibonacci s Birds from two towers Problem On a certain ground there are two towers, one of which is 30 feet high, the other 40, and they are only 50 feet apart; two birds descending together from the heights of the two towers fly to the center of a fountain between the towers; the distances from the center to both towers are sought. In the picture below, the taller tower is represented by the line segment BA, the shorter by DG. The center of the fountain is at Z. Since both birds start at the same time, and fly at the same speed, the segments GZ and AZ have the same length. One has to find the lengths of ZD and ZB. Solution. Let us write x = DZ (the length of DZ) and y = ZB we know that x + y = 50, so y = 50 x. Let h be the length of GZ, the hypotenuse of the triangle DGZ. This is also the length of AZ, the hypotenuse of the triangle ABZ. By Pythagoras: Equating the two expressions for h 2 : h 2 = x 2 = x 2, h 2 = y 2 = (50 x) x 2 = (50 x) 2 We now use a very famous formula giving the square of a sum or difference of two terms (a sum or difference of two terms is known as a binomial), namely (a ± b) 2 = a 2 ± 2ab + b 2.
5 5 So (50 x) 2 = (50)x + x 2 = x + x 2. Returning to the last equation before this digression, x 2 = x + x 2 = x + x 2. We can now: Subtract x 2 from both sides, add 100x to both sides, subtract 900 from both sides; we are left with 100x = Dividing both sides by 100 we get x = 32, the length of ZD. The length of ZB is then y = 50 x = A Tournament problem Fibonacci took part in a mathematical tournament where he solved three problems. Here is the third one. Three men possess a pile of money. One man owns 1/2 of the pile, another one owns 1/3 and the third man owns 1/6 of the pile. Each man takes some money from the pile, until nothing is left. But then the first man returns 1/2 of what he took, the second man returns 1/3 of what he took, and the third man returns 1/6 of what he took. If the returned total is divided evenly among the men, it is found that each then has what he is entitled to. How much money was in the original pile, and how much money did each man take? I am leaving this as a problem still to be solved.
Factoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More information5 th Grade Mathematics
5 th Grade Mathematics Instructional Week 20 Rectilinear area with fractional side lengths and realworld problems involving area and perimeter of 2dimensional shapes Paced Standards: 5.M.2: Find the
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationSQUARESQUARE ROOT AND CUBECUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationSection 2.3: Quadratic Equations
SECTION.3 Quadratic Equations Section.3: Quadratic Equations Solving by Factoring Solving by Completing the Square Solving by the Quadratic Formula The Discriminant Solving by Factoring MATH 1310 College
More informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More informationAssignment 5  Due Friday March 6
Assignment 5  Due Friday March 6 (1) Discovering Fibonacci Relationships By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1 ) 2 + (F n ) 2 that is, a
More informationChapter 8. Quadratic Equations and Functions
Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationREVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95
REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets
More informationMultiplication of Whole Numbers & Factors
Lesson Number 3 Multiplication of Whole Numbers & Factors Professor Weissman s Algebra Classroom Martin Weissman, Jonathan S. Weissman. & Tamara Farber What Are The Different Ways To Show Multiplication?
More informationSolutions to Exercises, Section 5.2
Instructor s Solutions Manual, Section 5.2 Exercise 1 Solutions to Exercises, Section 5.2 In Exercises 1 8, convert each angle to radians. 1. 15 Start with the equation Divide both sides by 360 to obtain
More informationThis module makes a connection between geometry and number.
SHOW 119 PROGRAM SYNOPSIS Segment 1 (5:55) CALLOUS CANDY BOX In a parody of the TV show Dallas, the scion of the Callous Candy Company proposes to commemorate the firm s 101st anniversary with a special
More informationACTIVITY: Multiplying Binomials Using Algebra Tiles. Work with a partner. Six different algebra tiles are shown below.
7.3 Multiplying Polynomials How can you multiply two binomials? 1 ACTIVITY: Multiplying Binomials Using Algebra Tiles Work with a partner. Six different algebra tiles are shown below. 1 1 x x x x Write
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationExample 1: If the sum of seven and a number is multiplied by four, the result is 76. Find the number.
EXERCISE SET 2.3 DUE DATE: STUDENT INSTRUCTOR: 2.3 MORE APPLICATIONS OF LINEAR EQUATIONS Here are a couple of reminders you may need for this section: perimeter is the distance around the outside of the
More informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More informationPark Forest Math Team. Meet #5. Algebra. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
More informationPythagorean Theorem. Inquiry Based Unit Plan
Pythagorean Theorem Inquiry Based Unit Plan By: Renee Carey Grade: 8 Time: 5 days Tools: Geoboards, Calculators, Computers (Geometer s Sketchpad), Overhead projector, Pythagorean squares and triangle manipulatives,
More informationcalled and explain why it cannot be factored with algebra tiles? and explain why it cannot be factored with algebra tiles?
Factoring Reporting Category Topic Expressions and Operations Factoring polynomials Primary SOL A.2c The student will perform operations on polynomials, including factoring completely first and seconddegree
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More information1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft
2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite
More informationThe Fibonacci Sequence and the Golden Ratio
55 The solution of Fibonacci s rabbit problem is examined in Chapter, pages The Fibonacci Sequence and the Golden Ratio The Fibonacci Sequence One of the most famous problems in elementary mathematics
More information1 Math 116 Supplemental Textbook (Pythagorean Theorem)
1 Math 116 Supplemental Textbook (Pythagorean Theorem) 1.1 Pythagorean Theorem 1.1.1 Right Triangles Before we begin to study the Pythagorean Theorem, let s discuss some facts about right triangles. The
More informationUnit 10: Quadratic Equations Chapter Test
Unit 10: Quadratic Equations Chapter Test Part 1: Multiple Choice. Circle the correct answer. 1. The area of a square is 169 cm 2. What is the length of one side of the square? A. 84.5 cm C. 42.25 cm B.
More informationSecond Grade Math Standards and I Can Statements
Second Grade Math Standards and I Can Statements Standard CC.2.OA.1 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationLaw of Sines. Definition of the Law of Sines:
Law of Sines So far we have been using the trigonometric functions to solve right triangles. However, what happens when the triangle does not have a right angle? When solving oblique triangles we cannot
More information9.6 The Pythagorean Theorem
Section 9.6 The Pythagorean Theorem 959 9.6 The Pythagorean Theorem Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in
More information10 7, 8. 2. 6x + 30x + 36 SOLUTION: 89 Perfect Squares. The first term is not a perfect square. So, 6x + 30x + 36 is not a perfect square trinomial.
Squares Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1.5x + 60x + 36 SOLUTION: The first term is a perfect square. 5x = (5x) The last term is a perfect
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
5 th GRADE Archdiocese of Washington Catholic Schools Standard 1  Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,
More informationRight Triangles and Quadrilaterals
CHATER. RIGHT TRIANGLE AND UADRILATERAL 18 1 5 11 Choose always the way that seems the best, however rough it may be; custom will soon render it easy and agreeable. ythagoras CHATER Right Triangles and
More informationMAT 080Algebra II Applications of Quadratic Equations
MAT 080Algebra II Applications of Quadratic Equations Objectives a Applications involving rectangles b Applications involving right triangles a Applications involving rectangles One of the common applications
More informationPythagorean Theorem: Proof and Applications
Pythagorean Theorem: Proof and Applications Kamel AlKhaled & Ameen Alawneh Department of Mathematics and Statistics, Jordan University of Science and Technology IRBID 22110, JORDAN Email: kamel@just.edu.jo,
More informationAdditional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman
Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman Email address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents
More informationPythagoras. 1 of 60. (Pythagoras)
Pythagoras 1 of 60 http://www.youtube.com/watch?v=8fjlxrudhg4 (Pythagoras) 2 of 60 The history of Pythagoras Theorem The theorem is named after the Greek mathematician and philosopher, Pythagoras. He lived
More informationfor the Common Core State Standards 2012
A Correlation of for the Common Core State s 2012 to the Common Core Georgia Performance s Grade 2 FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: K12 Mathematics
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationA BRIEF GUIDE TO ABSOLUTE VALUE. For HighSchool Students
1 A BRIEF GUIDE TO ABSOLUTE VALUE For HighSchool Students This material is based on: THINKING MATHEMATICS! Volume 4: Functions and Their Graphs Chapter 1 and Chapter 3 CONTENTS Absolute Value as Distance
More informationSolution Guide for Chapter 6: The Geometry of Right Triangles
Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More information78 Multiplying Polynomials
78 Multiplying Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationThe Product Property of Square Roots states: For any real numbers a and b, where a 0 and b 0, ab = a b.
Chapter 9. Simplify Radical Expressions Any term under a radical sign is called a radical or a square root expression. The number or expression under the the radical sign is called the radicand. The radicand
More informationMATH122 Final Exam Review If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71.
MATH1 Final Exam Review 5. 1. If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71. 5.1. If 8 secθ = and θ is an acute angle, find sinθ exactly. 3 5.1 3a. Convert
More informationGeometry FSA Mathematics Practice Test Questions
Geometry FSA Mathematics Practice Test Questions The purpose of these practice test materials is to orient teachers and students to the types of questions on paperbased FSA tests. By using these materials,
More informationGrade 6 Math Circles March 24/25, 2015 Pythagorean Theorem Solutions
Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 4/5, 015 Pythagorean Theorem Solutions Triangles: They re Alright When They
More information1.2. Successive Differences
1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers
More informationRectangle Square Triangle
HFCC Math Lab Beginning Algebra  15 PERIMETER WORD PROBLEMS The perimeter of a plane geometric figure is the sum of the lengths of its sides. In this handout, we will deal with perimeter problems involving
More informationUtah Core Curriculum for Mathematics
Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions
More information3. Find the yintercept of the function: f(x) = 6 x A wheelchair ramp runs 36 inches and rises 3 inches. What is the rate of change?
Coordinate Algebra EOCT Review Test #2 1. 4. For the function f(x) = 2x 2, if the domain is {3, 0, 3}, find the range. 2. An arithmetic sequence is: 5. The graph shows the first quadrant portion of four
More informationGeometry: A Better Understanding of Area
Geometry: A Better Understanding of Area 6G1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other
More informationSect 6.7  Solving Equations Using the Zero Product Rule
Sect 6.7  Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationMODERN APPLICATIONS OF PYTHAGORAS S THEOREM
UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented
More informationName Date Time. STUDY LINK 8 13 Unit 9: Family Letter. Please keep this Family Letter for reference as your child works through Unit 9.
Name Date Time STUDY LINK Unit 9: Family Letter More about Variables, Formulas, and Graphs You may be surprised at some of the topics that are covered in Unit 9. Several of them would be traditionally
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationUnit 6 Algebraic Investigations: Quadratics and More, Part 2
Accelerated Mathematics I Frameworks Student Edition Unit 6 Algebraic Investigations: Quadratics and More, Part nd Edition March, 011 Accelerated Mathematics I Unit 6 nd Edition Table of Contents INTRODUCTION:...
More informationSample Problems. 3. Find the missing leg of the right triangle shown on the picture below.
Lecture Notes The Pythagorean Theorem page 1 Sample Problems 1. Could the three line segments given below be the three sides of a right triangle? Explain your answer. a) 6 cm; 10 cm; and 8 cm b) 7 ft,
More informationMATH EXPRESSIONS GRADE 1  SCOPE AND SEQUENCE
UNIT 1: EARLY NUMBER ACTIVITIES Math Expressions, Investigations 2 Investigations Games: Dot Addition, Compare, Double Compare; On & Off; Counters in a Cup 1520 Days Apply properties of operations and
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationName: Class: Date: Geometry Chapter 3 Review
Name: Class: Date: ID: A Geometry Chapter 3 Review. 1. The area of a rectangular field is 6800 square meters. If the width of the field is 80 meters, what is the perimeter of the field? Draw a diagram
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More information5 th Grade Math. ELG 5.MD.C Understand concepts of volume and relate volume to multiplication and to addition. Vertical Progression: 3 rd Grade
Vertical Progression: 3 rd Grade 4 th Grade 5 th Grade 3.MD.C Geometric measurement: understanding concepts of area and relate area to multiplication and to addition. o 3.MD.C.5 Recognize area as an attribute
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationApplications of right triangles and trigonometry
Applications of right triangles and trigonometry 1. A sailor at sea in the 18 th century had no GPS or electronic navigational tools. The main tool used to determine the position of the ship was the sextant,
More informationGeorgia Department of Education Common Core Georgia Performance Standards Framework Fourth Grade Mathematics Unit
Fourth Grade Mathematics Unit Constructing Task: Area and Perimeter Adapted from Fixed Perimeters and Fixed Areas in Teaching StudentCentered MATHEMATICS Grades 35 by John Van de Walle and LouAnn Lovin.
More informationSection 1.4. Difference Equations
Difference Equations to Differential Equations Section 1.4 Difference Equations At this point almost all of our sequences have had explicit formulas for their terms. That is, we have looked mainly at sequences
More information1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.
1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was $10, but then it was reduced The number of customers
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationSect 8.3 Quadrilaterals, Perimeter, and Area
186 Sect 8.3 Quadrilaterals, Perimeter, and Area Objective a: Quadrilaterals Parallelogram Rectangle Square Rhombus Trapezoid A B E F I J M N Q R C D AB CD AC BD AB = CD AC = BD m A = m D m B = m C G H
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationMAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL
MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL THESE FUN MATH GAME PRINTABLES are brought to you with compliments from Making Math More Fun at and Math Board Games at Copyright
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationLinearizing Data. Lesson3. United States Population
Lesson3 Linearizing Data You may have heard that the population of the United States is increasing exponentially. The table and plot below give the population of the United States in the census years 19
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationChapter 5: Quadratic Function. What you ll learn. Chapter 5.1: Graphing Quadratic Functions. What you should learn. Graphing a Quadratic Function:
Chapter 5: Quadratic Function What you ll learn Chapter 5.1: Graphing Quadratic Functions What you should learn Graphing a Quadratic Function: Graphing Calculator activity, P249 The graph of a Quadratic
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationMathematics. Mathematical Practices
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with
More informationGrade 1 Math Expressions Vocabulary Words 2011
Italicized words Indicates OSPI Standards Vocabulary as of 10/1/09 Link to Math Expression Online Glossary for some definitions: http://wwwk6.thinkcentral.com/content/hsp/math/hspmathmx/na/gr1/se_9780547153179
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationLesson 4 Annuities: The Mathematics of Regular Payments
Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More information21 Creating and Solving Equations. I will create equations to represent situations and solve them to work out problems in context.
21 Creating and Solving Equations I will create equations to represent situations and solve them to work out problems in context. Linear Equations To create an equation to represent a word problem, follow
More informationTeacher Notes. Exploration: The Determinant of a Matrix. Learning outcomes addressed. Lesson Context. Lesson Launch.
Exploration Teacher Notes Exploration: The Determinant of a Matrix Learning outcomes addressed 3.7 Calculate the determinant of a matrix. 3.8 Evaluate a determinant to calculate the area of a triangle
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationDivision of Special Education LAUSD, June Grade 4 Math 156
Use the four operations with whole numbers to solve problems. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as
More informationGeorgia Department of Education THIRD GRADE MATHEMATICS UNIT 3 STANDARDS
Dear Parents, THIRD GRADE MATHEMATICS UNIT 3 STANDARDS We want to make sure that you have an understanding of the mathematics your child will be learning this year. Below you will find the standards we
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationAlgorithm set of steps used to solve a mathematical computation. Area The number of square units that covers a shape or figure
Fifth Grade CCSS Math Vocabulary Word List *Terms with an asterisk are meant for teacher knowledge only students need to learn the concept but not necessarily the term. Addend Any number being added Algorithm
More informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
More information1) Find the circumference and area of a circle with diameter of 15.8 in? (7.GM.5)
7.GM Unit 4 Practice Test 1) Find the circumference and area of a circle with diameter of 15.8 in? (7.GM.5) 2) What are the characteristics of each type of angle of adjacent, complementary, supplementary
More information