Location Quotients Location Quotients: a Tool for Comparing 1 Regional Industry Compositions a Tool for Comparing Regional Industry Compositions
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Elementary Number Theory and Methods of Proof
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright © Cengage Learning. All rights reserved. SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Copyright © Cengage Learning. All rights reserved. Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem The quotient-remainder theorem says that when any integer n is divided by any positive integer d, the result is a quotient q and a nonnegative remainder r that is smaller than d. 3 Example 1 – The Quotient-Remainder Theorem For each of the following values of n and d, find integers q and r such that and a. n = 54, d = 4 b. n = –54, d = 4 c. n = 54, d = 70 Solution: a. b. c. 4 div and mod 5 div and mod A number of computer languages have built-in functions that enable you to compute many values of q and r for the quotient-remainder theorem. These functions are called div and mod in Pascal, are called / and % in C and C++, are called / and % in Java, and are called / (or \) and mod in .NET. The functions give the values that satisfy the quotient-remainder theorem when a nonnegative integer n is divided by a positive integer d and the result is assigned to an integer variable. 6 div and mod However, they do not give the values that satisfy the quotient-remainder theorem when a negative integer n is divided by a positive integer d. 7 div and mod For instance, to compute n div d for a nonnegative integer n and a positive integer d, you just divide n by d and ignore the part of the answer to the right of the decimal point. -
Introduction to Uncertainties (Prepared for Physics 15 and 17)
Introduction to Uncertainties (prepared for physics 15 and 17) Average deviation. When you have repeated the same measurement several times, common sense suggests that your “best” result is the average value of the numbers. We still need to know how “good” this average value is. One measure is called the average deviation. The average deviation or “RMS deviation” of a data set is the average value of the absolute value of the differences between the individual data numbers and the average of the data set. For example if the average is 23.5cm/s, and the average deviation is 0.7cm/s, then the number can be expressed as (23.5 ± 0.7) cm/sec. Rule 0. Numerical and fractional uncertainties. The uncertainty in a quantity can be expressed in numerical or fractional forms. Thus in the above example, ± 0.7 cm/sec is a numerical uncertainty, but we could also express it as ± 2.98% , which is a fraction or %. (Remember, %’s are hundredths.) Rule 1. Addition and subtraction. If you are adding or subtracting two uncertain numbers, then the numerical uncertainty of the sum or difference is the sum of the numerical uncertainties of the two numbers. For example, if A = 3.4± .5 m and B = 6.3± .2 m, then A+B = 9.7± .7 m , and A- B = - 2.9± .7 m. Notice that the numerical uncertainty is the same in these two cases, but the fractional uncertainty is very different. Rule2. Multiplication and division. If you are multiplying or dividing two uncertain numbers, then the fractional uncertainty of the product or quotient is the sum of the fractional uncertainties of the two numbers. -
Unit 6: Multiply & Divide Fractions Key Words to Know
Unit 6: Multiply & Divide Fractions Learning Targets: LT 1: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b) LT 2: Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g. by using visual fraction models or equations to represent the problem. LT 3: Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction. LT 4: Interpret the product (a/b) q ÷ b. LT 5: Use a visual fraction model. Conversation Starters: Key Words § How are fractions like division problems? (for to Know example: If 9 people want to shar a 50-lb *Fraction sack of rice equally by *Numerator weight, how many pounds *Denominator of rice should each *Mixed number *Improper fraction person get?) *Product § 3 pizzas at 10 slices each *Equation need to be divided by 14 *Division friends. How many pieces would each friend receive? § How can a model help us make sense of a problem? Fractions as Division Students will interpret a fraction as division of the numerator by the { denominator. } What does a fraction as division look like? How can I support this Important Steps strategy at home? - Frac&ons are another way to Practice show division. https://www.khanacademy.org/math/cc- - Fractions are equal size pieces of a fifth-grade-math/cc-5th-fractions-topic/ whole. tcc-5th-fractions-as-division/v/fractions- - The numerator becomes the as-division dividend and the denominator becomes the divisor. Quotient as a Fraction Students will solve real world problems by dividing whole numbers that have a quotient resulting in a fraction. -
RATIO and PERCENT Grade Level: Fifth Grade Written By: Susan Pope, Bean Elementary, Lubbock, TX Length of Unit: Two/Three Weeks
RATIO AND PERCENT Grade Level: Fifth Grade Written by: Susan Pope, Bean Elementary, Lubbock, TX Length of Unit: Two/Three Weeks I. ABSTRACT A. This unit introduces the relationships in ratios and percentages as found in the Fifth Grade section of the Core Knowledge Sequence. This study will include the relationship between percentages to fractions and decimals. Finally, this study will include finding averages and compiling data into various graphs. II. OVERVIEW A. Concept Objectives for this unit: 1. Students will understand and apply basic and advanced properties of the concept of ratios and percents. 2. Students will understand the general nature and uses of mathematics. B. Content from the Core Knowledge Sequence: 1. Ratio and Percent a. Ratio (p. 123) • determine and express simple ratios, • use ratio to create a simple scale drawing. • Ratio and rate: solve problems on speed as a ratio, using formula S = D/T (or D = R x T). b. Percent (p. 123) • recognize the percent sign (%) and understand percent as “per hundred” • express equivalences between fractions, decimals, and percents, and know common equivalences: 1/10 = 10% ¼ = 25% ½ = 50% ¾ = 75% find the given percent of a number. C. Skill Objectives 1. Mathematics a. Compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators b. Use models to relate decimals to fractions that name tenths, hundredths, and thousandths c. Use fractions to describe the results of an experiment d. Use experimental results to make predictions e. Use table of related number pairs to make predictions f. Graph a given set of data using an appropriate graphical representation such as a picture or line g. -
4Fractions and Percentages
Fractions and Chapter 4 percentages What Australian you will learn curriculum 4A What are fractions? NUMBER AND ALGEBRA (Consolidating) 1 Real numbers 4B Equivalent fractions and Compare fractions using equivalence. Locate and represent fractions and simplifi ed fractions mixed numerals on a number line (ACMNA152) 4C Mixed numbers Solve problems involving addition and subtraction of fractions, including (Consolidating) those with unrelated denominators (ACMNA153) 4D Ordering fractions Multiply and divide fractions and decimals using ef cient written 4E Adding fractions strategies and digital technologies (ACMNA154) 4F Subtracting fractions Express one quantity as a fraction of another with and without the use of UNCORRECTEDdigital technologies (ACMNA155) 4G Multiplying fractions 4H Dividing fractions Connect fractions, decimals and percentages and carry out simple 4I Fractions and percentages conversions (ACMNA157) 4J Percentage of a number Find percentages of quantities and express one quantity as a percentage 4K Expressing aSAMPLE quantity as a of another, with and without digital PAGES technologies. (ACMNA158) proportion Recognise and solve problems involving simple ratios. (ACMNA173) 2 Money and nancial mathematics Investigate and calculate ‘best16x16 buys’, with and without digital32x32 technologies (ACMNA174) Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56882-2 • Ph 03 8671 1400 COP_04.indd 2 16/04/15 11:34 AM Online 173 resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses Ancient Egyptian fractions 1 The ancient Egyptians used fractions over 4000 years thought (eyebrow closest to brain) 8 ago. The Egyptian sky god Horus was a falcon-headed 1 hearing (pointing to ear) man whose eyes were believed to have magical healing 16 powers. -
Continued Fractions: Properties and Invariant Measure (Kettingbreuken: Eigenschappen En Invariante Maat)
Delft University of Technology Faculty Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Continued fractions: Properties and invariant measure (Kettingbreuken: Eigenschappen en invariante maat) Bachelor of science thesis by Angelina C. Lemmens Delft, Nederland 14 februari 2014 Inhoudsopgave 1 Introduction 1 2 Continued fractions 2 2.1 Regular continued fractions . .2 2.1.1 Convergence of the regular continued fraction . .4 2.2 N-expansion . .9 2.3 a/b-continued fractions . 11 3 Invariant measures 12 3.1 Definition of invariant measure . 12 3.2 Invariant measure of the regular continued fraction . 13 4 The Ergodic Theorem 15 4.1 Adler's Folklore Theorem . 15 4.2 Ergodicity of the regular continued fraction . 17 5 A new continued fraction 19 5.1 Convergence of the new continued fraction . 20 5.2 Invariant measure and ergodicity of the new continued fraction . 25 5.3 Simulation of the density of the new continued fraction . 28 6 Appendix 30 1 Introduction An irrational number can be represented in many ways. For example the repre- sentation could be a rational approximation and could be more convenient to work with than the original number. Representations can be for example the decimal expansion or the binary expansion. These representations are used in everyday life. Less known is the representation by continued fractions. Continued fractions give the best approximation of irrational numbers by rational numbers. This is something that Christiaan Huygens already knew (and used to construct his planetarium). Number representations like the decimal expansion and continued fractions have relations with probability theory and dynamical systems. -
Maths Revision & Practice Booklet
Maths Revision & Practice Booklet Name: Fractions, Decimals and Percentages visit twinkl.com Maths Revision & Practice Booklet Revise Using Common Factors to Simplify Fractions Fractions that have the same value but represent this using different denominators and numerators are equivalent. We can recognise and find equivalent fractions by multiplying or dividing the numerator and denominator by the same amount. When we simplify a fraction, we use the highest common factor of the numerator and denominator to reduce the fraction to the lowest term equivalent fraction (simplest form). Factors of 21: 1 3 7 The highest 21 common factor 36 Factors of 36: 1 2 3 4 6 9 12 18 36 is 3. 21 ÷ 3 = 7 7 36 ÷ 3 = 12 12 Using Common Multiples to Express Fractions in the Same Denomination To compare or calculate with fractions, we often need to give them a common denominator. We do this by looking at the denominators and finding their lowest common multiple. 3 4 5 × 7 = 35 5 7 35 is the lowest common multiple. Remember that whatever we do 3 3 × 7 = 21 4 4 × 5 = 20 to the denominator, we have to 5 × 7 = 35 7 × 5 = 35 5 7 do to the numerator. Page 2 of 11 visit twinkl.com Maths Revision & Practice Booklet Comparing and Ordering Fractions, Including Fractions > 1 To compare and order ...look at the fractions with the same 5 2 numerators. denominators… 9 > 9 …change the 5 3 fractions into and To compare and order 9 5 equivalent fractions with different 5 9 fractions with the × × denominators… lowest common 25 27 denominator. -
Division Into Cases and the Quotient-Remainder Theorem
4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem 4.4 Quotient-Remainder Theorem 1 / 4 1 n = 34 and d = 6. 2 n = −34 and d = 6. Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: 4.4 Quotient-Remainder Theorem 2 / 4 2 n = −34 and d = 6. The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. 1 n = 34 and d = 6. 4.4 Quotient-Remainder Theorem 2 / 4 The Quotient-Remainder Theorem Theorem Given any integer n and positive integer d, there exist unique integers q and r such that n = dq + r and 0 ≤ r < d: Examples For each of the following values of n and d, find integers q and r such that n = dq + r and 0 ≤ r < d. 1 n = 34 and d = 6. 2 n = −34 and d = 6. 4.4 Quotient-Remainder Theorem 2 / 4 1 Compute 33 div 9 and 33 mod 9 (by hand and Python). 2 Keeping in mind which years are leap years, what day of the week will be 1 year from today? 3 Suppose that m is an integer. -
Eureka Math™ Tips for Parents Module 2
Grade 6 Eureka Math™ Tips for Parents Module 2 The chart below shows the relationships between various fractions and may be a great Key Words tool for your child throughout this module. Greatest Common Factor In this 19-lesson module, students complete The greatest common factor of two whole numbers (not both zero) is the their understanding of the four operations as they study division of whole numbers, division greatest whole number that is a by a fraction, division of decimals and factor of each number. For operations on multi-digit decimals. This example, the GCF of 24 and 36 is 12 expanded understanding serves to complete because when all of the factors of their study of the four operations with positive 24 and 36 are listed, the largest rational numbers, preparing students for factor they share is 12. understanding, locating, and ordering negative rational numbers and working with algebraic Least Common Multiple expressions. The least common multiple of two whole numbers is the least whole number greater than zero that is a What Came Before this Module: multiple of each number. For Below is an example of how a fraction bar model can be used to represent the Students added, subtracted, and example, the LCM of 4 and 6 is 12 quotient in a division problem. multiplied fractions and decimals (to because when the multiples of 4 and the hundredths place). They divided a 6 are listed, the smallest or first unit fraction by a non-zero whole multiple they share is 12. number as well as divided a whole number by a unit fraction. -
Maths Module 3 Ratio, Proportion and Percent
Maths Module 3 Ratio, Proportion and Percent This module covers concepts such as: • ratio • direct and indirect proportion • rates • percent www.jcu.edu.au/students/learning-centre Module 3 Ratio, Proportion and Percent 1. Ratio 2. Direct and Indirect Proportion 3. Rate 4. Nursing Examples 5. Percent 6. Combine Concepts: A Word Problem 7. Answers 8. Helpful Websites 2 1. Ratio Understanding ratio is very closely related to fractions. With ratio comparisons are made between equal sized parts/units. Ratios use the symbol “ : “ to separate the quantities being compared. For example, 1:3 means 1 unit to 3 units, and the units are the same size. Whilst ratio can be expressed as a fraction, the ratio 1:3 is NOT the same as , as the rectangle below illustrates 1 3 The rectangle is 1 part grey to 3 parts white. Ratio is 1:3 (4 parts/units) The rectangle is a total of 4 parts, and therefore, the grey part represented symbolically is and not 1 1 4 3 AN EXAMPLE TO BEGIN: My two-stroke mower requires petrol and oil mixed to a ratio of 1:25. This means that I add one part oil to 25 parts petrol. No matter what measuring device I use, the ratio must stay the same. So if I add 200mL of oil to my tin, I add 200mL x 25 = 5000mL of petrol. Note that the total volume (oil and petrol combined) = 5200mL which can be converted to 5.2 litres. Ratio relationships are multiplicative. Mathematically: 1:25 is the same as 2:50 is the same as 100:2500 and so on. -
5 Working with Proportionality and Ratio
5 Working with proportionality and ratio Key words: proportional, directly proportional, line graph, origin, gradient, slope, horizontal axis, vertical axis, x-axis, y-axis, x-coordinate, y-coordinate, rate, constant, constant of proportionality, reciprocal, inverse, inversely proportional, ratio, percentage, scale, scale drawing, scale factor, linear dimension. There are many different kinds of relationship between variables. A very common relationship is when one variable is proportional to another and this section focuses on this kind of relationship. It also considers the related ideas of ratio, percentage and scale. 5.1 Meaning of proportional A formal way of expressing proportionality would be that variable A is proportional to variable B when the values of the two variables are related by a constant multiplier. It is easier to understand the idea of proportionality through an example. Banks count coins by weighing them. Suppose a coin has a mass of 5 g. Then two coins have a mass of 10 g. Doubling the number of coins doubles the mass. If there are 10 coins then they have 10 times the mass of one coin (i.e. 100 g). This is expressed by saying that the mass of coins is proportional to the number of coins. A proportional relationship also works the other way round. The number of coins is proportional to the mass of the coins. A bag with 100 g of coins contains 10 coins. If you have double the mass (200 g) then you have double the number of coins (20). This is essentially what the bank is doing when it weighs coins to count them. -
Hardin Middle School Math Cheat Sheets
Name: __________________________________________ 7th Grade Math Teacher: ______________________________ 8th Grade Math Teacher: ______________________________ Hardin Middle School Math Cheat Sheets You will be given only one of these books. If you lose the book, it will cost $5 to replace it. Compiled by Shirk &Harrigan - Updated May 2013 Alphabetized Topics Pages Pages Area 32 Place Value 9, 10 Circumference 32 Properties 12 22, 23, Comparing 23, 26 Proportions 24, 25, 26 Pythagorean Congruent Figures 35 36 Theorem 14, 15, 41 Converting 23 R.A.C.E. Divisibility Rules 8 Range 11 37, 38, 25 Equations 39 Rates Flow Charts Ratios 25, 26 32, 33, 10 Formulas 34 Rounding 18, 19, 20, 21, 35 Fractions 22, 23, Scale Factor 24 Geometric Figures 30, 31 Similar Figures 35 Greatest Common 22 Slide Method 22 Factor (GF or GCD) Inequalities 40 Substitution 29 Integers 18, 19 Surface Area 33 Ladder Method 22 Symbols 5 Least Common Multiple 22 Triangles 30, 36 (LCM or LCD) Mean 11 Variables 29 Median 11 Vocabulary Words 43 Mode 11 Volume 34 Multiplication Table 6 Word Problems 41, 42 Order of 16, 17 Operations 23, 24, Percent 25, 26 Perimeter 32 Table of Contents Pages Cheat Sheets 5 – 42 Math Symbols 5 Multiplication Table 6 Types of Numbers 7 Divisibility Rules 8 Place Value 9 Rounding & Comparing 10 Measures of Central Tendency 11 Properties 12 Coordinate Graphing 13 Measurement Conversions 14 Metric Conversions 15 Order of Operations 16 – 17 Integers 18 – 19 Fraction Operations 20 – 21 Ladder/Slide Method 22 Converting Fractions, Decimals, & 23 Percents Cross Products 24 Ratios, Rates, & Proportions 25 Comparing with Ratios, Percents, 26 and Proportions Solving Percent Problems 27 – 28 Substitution & Variables 29 Geometric Figures 30 – 31 Area, Perimeter, Circumference 32 Surface Area 33 Volume 34 Congruent & Similar Figures 35 Pythagorean Theorem 36 Hands-On-Equation 37 Understanding Flow Charts 38 Solving Equations Mathematically 39 Inequalities 40 R.A.C.E.