The document defines mixed numbers and improper fractions. A mixed number has two parts - a whole number and a fraction, while an improper fraction has a numerator larger than the denominator. The document provides steps for changing between mixed numbers and improper fractions, including multiplying the denominator by the whole number and adding it to the numerator.
Proper; Improper & Mixed Number FractionsLorenKnights
This document discusses different types of fractions:
- Proper fractions have a numerator less than the denominator (e.g. 1/4).
- Improper fractions have a numerator greater than or equal to the denominator (e.g. 5/3).
- Mixed numbers are a combination of a whole number and a proper fraction (e.g. 2 1/4).
The document provides examples of converting between improper fractions and mixed numbers by dividing the numerator by the denominator to get the whole number part and remainder.
This document discusses different methods for comparing fractions, including:
1) Comparing fractions with the same denominator by looking at the numerators
2) Making the denominators the same by finding the least common multiple before comparing
3) Comparing fractions by multiplying the numerators and denominators
4) Converting fractions to decimals and comparing the decimal forms
The key steps are to simplify the fractions to have a common denominator or convert to decimals before determining which fraction is greater.
This document provides an overview of fractions including:
- Definitions of proper, improper, and mixed numbers
- Equivalent fractions and how to identify them
- Ordering fractions with both like and unlike denominators
- Adding and subtracting fractions with both like and unlike denominators
- Examples are provided for each concept along with practice problems for students to work through
The document covers essential fraction concepts and provides clear explanations, examples, and practice problems to help students understand fractions.
Add Fractions With Unlike DenominatorsBrooke Young
This document provides steps for adding fractions with unlike denominators:
1) Find equivalent fractions with a common denominator
2) Add the numerators and use the sum as the new numerator
3) Keep the common denominator as the denominator
4) Simplify the resulting fraction if possible by reducing to lowest terms
Worked examples demonstrate applying the steps to add several pairs of fractions.
This document provides lessons and examples for multiplying mixed numbers. It begins with a warm up problem, then presents the concept of multiplying mixed numbers by first converting them to improper fractions. Several examples are worked through, showing how to multiply fractions and mixed numbers by multiplying corresponding numerators and denominators. Check problems are also included. The document ends with a short quiz assessing understanding of multiplying mixed numbers.
This document discusses comparing and ordering fractions. It provides examples of using less than, greater than, and equal to symbols to compare fractions with different denominators. It explains that to do this, you need to find the least common denominator and rewrite the fractions with equivalent denominators. The document also gives examples of ordering fractions from least to greatest value by finding a common denominator and comparing the numerators.
Converting fractions improper to mixed numbersDoreen Shottek
This document provides instructions for converting between mixed numbers and improper fractions. It defines mixed numbers as having a whole number part and fractional part, and improper fractions as fractions where the numerator is larger than the denominator. The document then provides step-by-step instructions for converting a mixed number to an improper fraction by multiplying the whole number by the denominator and adding the numerator, and for converting an improper fraction to a mixed number by dividing the numerator by the denominator and writing the remainder over the denominator. Several examples are worked through.
Proper; Improper & Mixed Number FractionsLorenKnights
This document discusses different types of fractions:
- Proper fractions have a numerator less than the denominator (e.g. 1/4).
- Improper fractions have a numerator greater than or equal to the denominator (e.g. 5/3).
- Mixed numbers are a combination of a whole number and a proper fraction (e.g. 2 1/4).
The document provides examples of converting between improper fractions and mixed numbers by dividing the numerator by the denominator to get the whole number part and remainder.
This document discusses different methods for comparing fractions, including:
1) Comparing fractions with the same denominator by looking at the numerators
2) Making the denominators the same by finding the least common multiple before comparing
3) Comparing fractions by multiplying the numerators and denominators
4) Converting fractions to decimals and comparing the decimal forms
The key steps are to simplify the fractions to have a common denominator or convert to decimals before determining which fraction is greater.
This document provides an overview of fractions including:
- Definitions of proper, improper, and mixed numbers
- Equivalent fractions and how to identify them
- Ordering fractions with both like and unlike denominators
- Adding and subtracting fractions with both like and unlike denominators
- Examples are provided for each concept along with practice problems for students to work through
The document covers essential fraction concepts and provides clear explanations, examples, and practice problems to help students understand fractions.
Add Fractions With Unlike DenominatorsBrooke Young
This document provides steps for adding fractions with unlike denominators:
1) Find equivalent fractions with a common denominator
2) Add the numerators and use the sum as the new numerator
3) Keep the common denominator as the denominator
4) Simplify the resulting fraction if possible by reducing to lowest terms
Worked examples demonstrate applying the steps to add several pairs of fractions.
This document provides lessons and examples for multiplying mixed numbers. It begins with a warm up problem, then presents the concept of multiplying mixed numbers by first converting them to improper fractions. Several examples are worked through, showing how to multiply fractions and mixed numbers by multiplying corresponding numerators and denominators. Check problems are also included. The document ends with a short quiz assessing understanding of multiplying mixed numbers.
This document discusses comparing and ordering fractions. It provides examples of using less than, greater than, and equal to symbols to compare fractions with different denominators. It explains that to do this, you need to find the least common denominator and rewrite the fractions with equivalent denominators. The document also gives examples of ordering fractions from least to greatest value by finding a common denominator and comparing the numerators.
Converting fractions improper to mixed numbersDoreen Shottek
This document provides instructions for converting between mixed numbers and improper fractions. It defines mixed numbers as having a whole number part and fractional part, and improper fractions as fractions where the numerator is larger than the denominator. The document then provides step-by-step instructions for converting a mixed number to an improper fraction by multiplying the whole number by the denominator and adding the numerator, and for converting an improper fraction to a mixed number by dividing the numerator by the denominator and writing the remainder over the denominator. Several examples are worked through.
The document discusses mixed numbers and how to add and convert fractions. It explains that a mixed number has a whole number part and fractional part. It provides examples of finding common denominators and equivalent fractions to add fractions with different denominators. It demonstrates how to convert improper fractions to mixed numbers.
The document discusses how to divide 4 jelly beans between 2 people. It explains key terms used in division such as dividend, divisor, and quotient. It then provides examples of dividing numbers by 1, 0, and themselves. The document outlines different methods for division, including repeated subtraction, using objects to demonstrate groups, and the horizontal and long division methods. It also provides examples of dividing multiples of 10, 100, and 1000 by those same numbers.
Helping parents to understand the correct method of teaching their children Algebra / Mathematics / Math can be tricky.
There are many pit-falls in helping children with their homework because many of the ways we were taught are out of date.
Try this simple free online lesson and watch as your child learns how to do Simple Division by following this step-by-step guide.
The document discusses adding and subtracting fractions with like denominators. It provides steps for adding and subtracting numerators while keeping the denominators the same. Examples are shown of adding and subtracting fractions with like denominators. Additional practice problems are presented for the reader to work through.
Multiplication is a repeated addition. It can be represented by using fingers to count groups being added together. The order of the factors does not change the product, so 2 x 3 equals 3 x 2 and both equal 6. Practice multiplication by representing problems with fingers to find that the product is the same regardless of the order of the factors.
This document provides instructions for multiplying, dividing, and converting between decimals and fractions. It explains the basic steps:
1) Count the decimal places and line up the numbers accordingly.
2) Ignore the decimal point and multiply or divide the numbers as whole numbers.
3) Place the decimal point by counting places left or right from the original decimal.
Several examples are worked through, such as multiplying 5.8 x 7 and dividing 21.086 by 3. Then readers are prompted to try problems themselves, like dividing 0.4 by 0.0025.
This document provides steps for subtracting fractions, including similar fractions, dissimilar fractions, and mixed numbers. For similar fractions, subtract the numerators and keep the denominator. For dissimilar fractions, find the lowest common denominator (LCD) and convert the fractions before subtracting. For mixed numbers, change the fractions to similar fractions using the LCD, then subtract the whole numbers and numerators while keeping the denominator. Simplify the final answer if possible by expressing it in lowest terms.
This document discusses factors and multiples of numbers. It defines factors as numbers that can be multiplied together to get another number, and multiples as numbers obtained by multiplying a given number by whole numbers. Examples are provided to illustrate factors, multiples, and types of numbers including abundant, deficient, and perfect numbers. Practice problems are included to identify factors and multiples, as well as grouping students into equal numbers of groups.
Adding Fractions with Unlike Denominatorsmrmitchell3
To add fractions with unlike denominators, you first find the least common denominator (LCD) and then convert the fractions to equivalent fractions with the LCD as the denominator. You then add the numerators and keep the LCD as the denominator of the final sum. For example, to add 2/5 + 1/3: find the LCD of 5 and 3 is 15; convert to equivalent fractions 6/15 + 5/15; add the numerators to get 11/15.
This document provides instructions on how to simplify fractions by dividing the numerator and denominator by the largest number that divides both. It includes examples of simplifying fractions through division and with a calculator. Key rules are explained, such as using the largest number that divides both the numerator and denominator to get the fraction into its lowest terms. Practice questions are provided to reinforce these concepts.
Long division is explained using a family as an analogy to represent the steps. Dad represents dividing, Mom represents multiplying, Sister represents subtracting, Brother represents bringing down digits, and Rover represents repeating the process or getting the remainder. The document then walks through a long division problem step-by-step using this personification analogy to illustrate each part of the long division process.
Common Factors And Greatest Common FactorBrooke Young
This document discusses finding the common factors and greatest common factor (GCF) of two numbers. It provides examples of finding the common factors and GCF of 40 and 45 (which is 5), 13 and 15 (which is 1), and 18 and 24 (which is 6). To find the GCF, you list all the factors of each number, identify the common factors, and from those choose the greatest value as the GCF. While there may be multiple common factors, there is only one GCF.
This document contains lesson materials on operations with fractions, including examples of addition, subtraction, multiplication, and division of fractions. It provides steps for solving each type of operation, such as multiplying the numerators and denominators for multiplication, or applying cross multiplication for division. It then includes practice problems for students to work through, covering adding and subtracting similar and dissimilar fractions, as well as multiplying and dividing fractions. The document aims to teach students the key steps and methods for performing different mathematical operations with fractions.
The document explains how multiplication and division are related. Multiplication is a shortcut for addition of equal groups, while division is the opposite of multiplication and involves splitting things into equal groups. Examples are provided to illustrate how to use multiplication to solve division problems by thinking of the division sign as asking "what number multiplied by the given number equals the total?"
The document defines key terms related to fractions such as numerator, denominator, improper fraction, proper fraction, and equivalent fractions. It provides examples of how to read fractions such as 1/2, 1/3, 1/4, and discusses that fractions represent parts of a whole. Fractions can also represent division of numbers as either terminating or recurring decimals.
This document provides instructions for converting decimals to fractions and fractions to decimals. It explains that the place value of the last digit determines the denominator of the fraction. For decimals, the place value is determined by powers of ten. For fractions, the place value determines where the digit goes in the decimal. It also addresses situations where the denominator is not a power of ten, in which case the fraction needs to be divided.
The document discusses the concept of the least common multiple (LCM). It defines the LCM as the lowest number that is a multiple of two or more numbers. It provides examples of finding the LCM of different pairs of numbers by listing their multiples and circling the first number that is common to both lists. The document also discusses how the LCM can be used to find patterns involving multiples and to add or subtract fractions by finding a common denominator.
Multiplying fractions involves multiplying the numerators and denominators, then simplifying the resulting fraction. When multiplying a fraction by a whole number, change the whole number to an equivalent fraction. It is often easier to simplify fractions before multiplying them. To multiply mixed numbers, first convert them to improper fractions, then multiply the numerators and denominators and simplify. Practice problems are provided for the learner to try.
A fraction is in lowest form when the greatest common factor (GCF) of the numerator and denominator is 1. To express a fraction in lowest form, you find the GCF of the numerator and denominator and divide both by the GCF. This reduces the fraction to a form where the numerator and denominator cannot be further divided.
The document is a summary of lecture notes for a Calculus I class. It discusses integration by substitution, providing theory, examples, and objectives. Key points covered include the substitution rule for indefinite integrals, working through examples like finding the integral of √x2+1 dx, and noting substitution can transform integrals into simpler forms. Definite integrals using substitution are also briefly mentioned.
The document discusses techniques for sketching graphs of functions, including:
- Using the increasing/decreasing test to determine if a function is increasing or decreasing based on the sign of the derivative
- Using the concavity test to determine if a graph is concave up or down based on the second derivative
- A checklist for completely graphing a function, including finding critical points, inflection points, asymptotes, and putting together the information about monotonicity and concavity.
The document discusses mixed numbers and how to add and convert fractions. It explains that a mixed number has a whole number part and fractional part. It provides examples of finding common denominators and equivalent fractions to add fractions with different denominators. It demonstrates how to convert improper fractions to mixed numbers.
The document discusses how to divide 4 jelly beans between 2 people. It explains key terms used in division such as dividend, divisor, and quotient. It then provides examples of dividing numbers by 1, 0, and themselves. The document outlines different methods for division, including repeated subtraction, using objects to demonstrate groups, and the horizontal and long division methods. It also provides examples of dividing multiples of 10, 100, and 1000 by those same numbers.
Helping parents to understand the correct method of teaching their children Algebra / Mathematics / Math can be tricky.
There are many pit-falls in helping children with their homework because many of the ways we were taught are out of date.
Try this simple free online lesson and watch as your child learns how to do Simple Division by following this step-by-step guide.
The document discusses adding and subtracting fractions with like denominators. It provides steps for adding and subtracting numerators while keeping the denominators the same. Examples are shown of adding and subtracting fractions with like denominators. Additional practice problems are presented for the reader to work through.
Multiplication is a repeated addition. It can be represented by using fingers to count groups being added together. The order of the factors does not change the product, so 2 x 3 equals 3 x 2 and both equal 6. Practice multiplication by representing problems with fingers to find that the product is the same regardless of the order of the factors.
This document provides instructions for multiplying, dividing, and converting between decimals and fractions. It explains the basic steps:
1) Count the decimal places and line up the numbers accordingly.
2) Ignore the decimal point and multiply or divide the numbers as whole numbers.
3) Place the decimal point by counting places left or right from the original decimal.
Several examples are worked through, such as multiplying 5.8 x 7 and dividing 21.086 by 3. Then readers are prompted to try problems themselves, like dividing 0.4 by 0.0025.
This document provides steps for subtracting fractions, including similar fractions, dissimilar fractions, and mixed numbers. For similar fractions, subtract the numerators and keep the denominator. For dissimilar fractions, find the lowest common denominator (LCD) and convert the fractions before subtracting. For mixed numbers, change the fractions to similar fractions using the LCD, then subtract the whole numbers and numerators while keeping the denominator. Simplify the final answer if possible by expressing it in lowest terms.
This document discusses factors and multiples of numbers. It defines factors as numbers that can be multiplied together to get another number, and multiples as numbers obtained by multiplying a given number by whole numbers. Examples are provided to illustrate factors, multiples, and types of numbers including abundant, deficient, and perfect numbers. Practice problems are included to identify factors and multiples, as well as grouping students into equal numbers of groups.
Adding Fractions with Unlike Denominatorsmrmitchell3
To add fractions with unlike denominators, you first find the least common denominator (LCD) and then convert the fractions to equivalent fractions with the LCD as the denominator. You then add the numerators and keep the LCD as the denominator of the final sum. For example, to add 2/5 + 1/3: find the LCD of 5 and 3 is 15; convert to equivalent fractions 6/15 + 5/15; add the numerators to get 11/15.
This document provides instructions on how to simplify fractions by dividing the numerator and denominator by the largest number that divides both. It includes examples of simplifying fractions through division and with a calculator. Key rules are explained, such as using the largest number that divides both the numerator and denominator to get the fraction into its lowest terms. Practice questions are provided to reinforce these concepts.
Long division is explained using a family as an analogy to represent the steps. Dad represents dividing, Mom represents multiplying, Sister represents subtracting, Brother represents bringing down digits, and Rover represents repeating the process or getting the remainder. The document then walks through a long division problem step-by-step using this personification analogy to illustrate each part of the long division process.
Common Factors And Greatest Common FactorBrooke Young
This document discusses finding the common factors and greatest common factor (GCF) of two numbers. It provides examples of finding the common factors and GCF of 40 and 45 (which is 5), 13 and 15 (which is 1), and 18 and 24 (which is 6). To find the GCF, you list all the factors of each number, identify the common factors, and from those choose the greatest value as the GCF. While there may be multiple common factors, there is only one GCF.
This document contains lesson materials on operations with fractions, including examples of addition, subtraction, multiplication, and division of fractions. It provides steps for solving each type of operation, such as multiplying the numerators and denominators for multiplication, or applying cross multiplication for division. It then includes practice problems for students to work through, covering adding and subtracting similar and dissimilar fractions, as well as multiplying and dividing fractions. The document aims to teach students the key steps and methods for performing different mathematical operations with fractions.
The document explains how multiplication and division are related. Multiplication is a shortcut for addition of equal groups, while division is the opposite of multiplication and involves splitting things into equal groups. Examples are provided to illustrate how to use multiplication to solve division problems by thinking of the division sign as asking "what number multiplied by the given number equals the total?"
The document defines key terms related to fractions such as numerator, denominator, improper fraction, proper fraction, and equivalent fractions. It provides examples of how to read fractions such as 1/2, 1/3, 1/4, and discusses that fractions represent parts of a whole. Fractions can also represent division of numbers as either terminating or recurring decimals.
This document provides instructions for converting decimals to fractions and fractions to decimals. It explains that the place value of the last digit determines the denominator of the fraction. For decimals, the place value is determined by powers of ten. For fractions, the place value determines where the digit goes in the decimal. It also addresses situations where the denominator is not a power of ten, in which case the fraction needs to be divided.
The document discusses the concept of the least common multiple (LCM). It defines the LCM as the lowest number that is a multiple of two or more numbers. It provides examples of finding the LCM of different pairs of numbers by listing their multiples and circling the first number that is common to both lists. The document also discusses how the LCM can be used to find patterns involving multiples and to add or subtract fractions by finding a common denominator.
Multiplying fractions involves multiplying the numerators and denominators, then simplifying the resulting fraction. When multiplying a fraction by a whole number, change the whole number to an equivalent fraction. It is often easier to simplify fractions before multiplying them. To multiply mixed numbers, first convert them to improper fractions, then multiply the numerators and denominators and simplify. Practice problems are provided for the learner to try.
A fraction is in lowest form when the greatest common factor (GCF) of the numerator and denominator is 1. To express a fraction in lowest form, you find the GCF of the numerator and denominator and divide both by the GCF. This reduces the fraction to a form where the numerator and denominator cannot be further divided.
The document is a summary of lecture notes for a Calculus I class. It discusses integration by substitution, providing theory, examples, and objectives. Key points covered include the substitution rule for indefinite integrals, working through examples like finding the integral of √x2+1 dx, and noting substitution can transform integrals into simpler forms. Definite integrals using substitution are also briefly mentioned.
The document discusses techniques for sketching graphs of functions, including:
- Using the increasing/decreasing test to determine if a function is increasing or decreasing based on the sign of the derivative
- Using the concavity test to determine if a graph is concave up or down based on the second derivative
- A checklist for completely graphing a function, including finding critical points, inflection points, asymptotes, and putting together the information about monotonicity and concavity.
Biology - Chp 1 - Biology The Study Of Life - PowerPointMel Anthony Pepito
1. The document provides instructions for students to complete a "Do Now" assignment at the beginning of class involving defining what makes something alive.
2. It then summarizes key concepts from Chapter 1 of a biology textbook including defining biology as the study of life and outlining characteristics of living things such as organization, reproduction, growth/development, response to stimuli, and evolution.
3. The methods used in biology like observation, hypothesis formation, experimentation, data analysis and theory development are described along with examples. Safety procedures and metric units are also covered.
The document contains a series of questions about identifying synonyms. It presents a word in bold and then 3 potential synonyms as answer choices to choose from. Some of the questions include identifying synonyms for words like "doubtful", "annual", "mandatory", "occasion", and "insolent". The purpose is to test the reader's knowledge of synonyms and ability to choose the word that means the same thing as the given word.
The document provides steps for evaluating expressions with variables. It works through examples of evaluating 6x, 6x3, 5n-3 for different values of n, 3.6/y + 2.8 for different values of y. For each example it shows substituting the value for the variable, then performing the calculations step-by-step according to the order of operations, providing the final solution.
1. Cells are the basic units of structure and function in living organisms. A microscope is needed to view cells because they are too small to be seen with the naked eye.
2. The basic parts of a cell include a cell membrane, cytoplasm, and organelles. Eukaryotic cells also contain a nucleus. Cells come in two basic types - prokaryotic cells lack a nucleus while eukaryotic cells contain a nucleus.
3. Cells are organized into tissues, organs, and organ systems to carry out specialized functions in living organisms.
1. The document discusses covalent bonding and molecular compounds. It defines covalent bonds as the sharing of electrons between nonmetal atoms.
2. Molecular compounds are formed from covalent bonds between atoms. They have lower melting and boiling points than ionic compounds.
3. Molecular formulas show the number and type of atoms in a molecule, but not their arrangement. Water's molecular formula is H2O.
This document contains notes from a chemistry chapter about matter and changes. It defines matter as anything that has mass and volume. It describes properties of matter as either extensive, depending on amount, or intensive, depending on type. The three states of matter are solids, liquids, and gases. Physical changes alter a material's properties without changing its composition and can be classified as reversible or irreversible. Mixtures contain two or more substances mixed together either homogeneously or heterogeneously. Solutions are homogeneous mixtures that can be separated using physical means like decanting, filtration, distillation, magnets, or sorting.
Chemistry - Chp 6 - The Periodic Table Revisited - PowerPointMel Anthony Pepito
The document discusses how elements are organized in the periodic table based on their atomic structure and properties, including trends in atomic size, ion formation, and other periodic properties as the atomic number increases and across periods. Early periodic tables organized just a few elements, but the modern periodic table is based on the periodic law and arranges all known elements in order of increasing atomic number and identifies trends in their physical and chemical properties.
The document discusses scientific measurement and units. It covers accuracy, precision, and significant figures when making measurements. Conversion factors allow measurements to be converted between different units through multiplication. Dimensional analysis uses the units of measurements to solve conversion problems by breaking them into steps. Complex problems are best solved by breaking them into manageable parts.
This document provides an overview of key concepts about matter and chemical changes from a chemistry textbook. It defines matter and its three main states (solid, liquid, gas). It describes properties as either extensive (depending on amount) or intensive (depending on type). It differentiates between physical and chemical changes, elements and compounds, and mixtures and pure substances. It outlines clues that indicate a chemical change has occurred and introduces the law of conservation of mass.
Chemistry - Chp 1 - Introduction To Chemistry - PowerPointMel Anthony Pepito
The document introduces chemistry by defining it as the study of matter, its properties, and the changes it undergoes, discussing the major areas of chemistry and how it relates to everyday life and future careers, and outlining the scientific method and approaches to solving problems, emphasizing developing a plan, performing calculations correctly for numeric problems, and applying concepts for conceptual problems.
This document discusses models of the atom and electron configuration. It begins by describing historical atomic models including Rutherford's model with a small, dense nucleus and electrons in orbits. Bohr's model improved on this by proposing electrons exist in specific energy levels. The modern quantum mechanical model describes electrons as probability clouds rather than definite orbits. The document then discusses electron configuration notation, including building up configurations using the aufbau principle and exceptions due to Hund's rule and half-filled orbitals. It concludes by introducing atomic spectra and the relationship between light and electron energy levels.
This document discusses properties of solutions and concentration of solutions. It covers factors that determine the rate and amount of solute that dissolves in a solvent, such as temperature, stirring, and surface area. The maximum amount of solute that can dissolve is called the solubility. Units used to express solubility include grams of solute per 100 grams of solvent. The concentration of a solution can be quantified using molarity, which is the moles of solute per liter of solution. More concentrated solutions have more solute dissolved in a smaller amount of solvent, while dilute solutions have less solute in a larger amount of solvent.
Fungi are eukaryotic heterotrophs that feed by absorbing nutrients from outside their bodies. They have cell walls containing chitin and their bodies are composed of filaments called hyphae that form a tangled mass called a mycelium. Fungi reproduce both sexually and asexually through spores. They play important ecological roles as decomposers that break down organic matter, parasites that infect plants and animals, and symbionts that form relationships like lichens and mycorrhizae.
The document summarizes key concepts from a chapter on the history of life as revealed by the fossil record. It describes how paleontologists study fossils to understand past life forms and environments, and how fossils provide evidence that life has changed over time with most species going extinct. It also discusses techniques for dating fossils, important patterns in macroevolution like extinction events, adaptive radiation and convergent evolution, and the role of developmental genes in shaping body plans over long periods of time.
This document provides an overview of ecology and key ecological concepts. It discusses the biosphere and levels of ecological organization from species to biomes. The document covers ecological methods of observation, experimentation and modeling. It then examines energy flow through ecosystems, including producers, consumers, trophic levels, and ecological pyramids. Finally, the document details the water, carbon, nitrogen, phosphorus and other nutrient cycles that allow for recycling of matter within ecosystems. Nutrient limitations can impact an ecosystem's primary productivity.
This chapter discusses methods for measuring quantities of substances. It introduces the mole as a unit for measuring amounts of matter equal to Avogadro's number of representative particles. The mole allows for easy conversion between mass and number of particles. Compound formulas allow calculating formula mass in grams per mole. Relationships between moles, mass, and volume at standard temperature and pressure enable interconversion between these quantities.
Chemistry - Chp 7 - Ionic and Metallic Bonding - PowerPointMel Anthony Pepito
This document summarizes key concepts from Chapter 7 on ionic and metallic bonding. It discusses how ions form via gaining or losing valence electrons to achieve stable noble gas configurations, and how ionic and metallic bonds differ. Ionic compounds are crystalline solids with high melting points that conduct when molten or dissolved. Metals have mobile valence electrons that allow conduction and properties like malleability. Alloys combine metals for improved properties.
Biology - Chp 2 - Hydrolysis And Dehydration Synthesis - PowerPointMel Anthony Pepito
Trypsin works best at basic pH levels while pepsin works best at acidic pH levels found in the stomach. Raising the stomach's pH would decrease pepsin's effectiveness at breaking down proteins. The document discusses how hydrolysis breaks down complex molecules like maltose into simpler molecules like glucose through the addition of water. Dehydration synthesis combines simpler molecules like glucose together to form more complex ones like maltose or starch by removing water.
This document discusses mixed numbers and improper fractions. It defines a mixed number as having a whole number part and fractional part, and an improper fraction as having a numerator larger than the denominator. It provides examples of changing between mixed numbers and improper fractions by multiplying the whole number by the denominator and adding the numerator, or dividing the numerator by the denominator and keeping the remainder over the denominator.
Kungfu math p4 slide7 (improper fractions)pdfkungfumath
This document discusses mixed numbers and improper fractions. It defines proper, improper, and mixed fractions. It provides examples of converting between improper and mixed fractions by multiplying or dividing the whole number by the denominator and writing any remainder over the denominator. Steps are given for converting mixed fractions to improper fractions by multiplying the whole number by the denominator and adding to the numerator, and for converting improper fractions to mixed fractions by dividing the numerator by the denominator.
The document provides information on operations with fractions, including:
1) Adding and subtracting fractions requires changing denominators to a common denominator and applying the operation to the numerators, keeping the denominator the same.
2) Subtracting a negative fraction moves the numerator to the right on the number line, making it larger, while adding a negative moves the numerator left, making it smaller.
3) Mixed numbers represent numbers that include both a whole number and a fraction, and can be changed to or from improper fractions through multiplication and addition/subtraction.
Converting Improper and Mixed Fractions.pptxJohdener14
This document discusses converting between improper fractions and mixed numbers. It provides examples of converting 7/3 to the mixed number 2 1/3, and converting the mixed number 1 2/4 to the improper fraction 9/2. The key steps shown are dividing the numerator by the denominator to get the whole number part of the mixed number, with the remainder becoming the numerator of the fractional part, and for converting a mixed number to an improper fraction, multiplying the denominator by the whole number and adding the numerator.
The document provides information on operations with fractions, including:
1) How to add and subtract fractions by finding a common denominator and then adding or subtracting the numerators, keeping the denominator the same.
2) How adding or subtracting a negative fraction affects the numerator on the number line.
3) Examples of adding and subtracting fractions.
4) The definition of a mixed number as a number that includes both a whole number and a fraction. Methods for converting between mixed numbers and improper fractions are also described.
This document explains how to change improper fractions to mixed fractions. An improper fraction has a numerator larger than or equal to the denominator, while a mixed fraction is a whole number and proper fraction combined. To change an improper fraction to a mixed fraction, divide the numerator by the denominator. The number of times the denominator divides into the numerator becomes the whole number part, and the remainder is the numerator of the fractional part over the original denominator. Examples are provided to demonstrate this process.
This document explains how to change improper fractions to mixed fractions. An improper fraction has a numerator larger than or equal to the denominator, while a mixed fraction is a whole number and proper fraction combined. To change an improper fraction to a mixed fraction, divide the numerator by the denominator. The number of times the denominator divides into the numerator becomes the whole number part, and the remainder is the numerator of the fractional part over the original denominator. Examples are provided to demonstrate this process.
This document discusses mixed numbers and improper fractions. An improper fraction has a numerator greater than the denominator, while a mixed number is a whole number with a proper fraction. The document provides steps to convert between mixed numbers and improper fractions, such as multiplying the whole number by the denominator and adding it to the numerator to convert a mixed number to an improper fraction.
This document provides information on various fraction concepts and operations including:
1) Adding similar fractions by adding the numerators and copying the denominators.
2) Multiplying fractions by multiplying the numerators and denominators.
3) Dividing fractions by changing the second fraction to its reciprocal and multiplying.
4) Performing operations on decimals by aligning the decimal points and applying the same rules as whole numbers.
The document outlines steps for working with fractions including: changing improper fractions to mixed numbers and vice versa; reducing fractions; raising fractions to higher terms; finding the part of one number that is another number; and comparing fractions. Key steps are to divide the numerator by the denominator to change an improper fraction to a mixed number, multiply the whole number by the denominator and add the numerator to change a mixed number to an improper fraction, and find a common denominator to compare fractions.
The document outlines steps for working with fractions including: changing improper fractions to mixed numbers and vice versa; reducing fractions; raising fractions to higher terms; finding what part one number is of another; and comparing fractions. Key steps are to divide the numerator by the denominator to change an improper fraction to a mixed number, multiply the whole number by the denominator and add the numerator to change a mixed number to an improper fraction, and find a common denominator to compare fractions.
An improper fraction has a numerator greater than the denominator, while a mixed number contains both a whole number and a fraction. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add it to the numerator, using the original denominator. To convert an improper fraction to a mixed number, divide the numerator by the denominator, using the quotient as the whole number and any remainder as the new numerator, keeping the original denominator.
This document discusses improper fractions, mixed numbers, and how to convert between the two. It defines an improper fraction as having a numerator larger than the denominator, and a mixed number as a whole number combined with a proper fraction. It provides examples and step-by-step instructions for converting a mixed number to an improper fraction by multiplying, adding, and writing the result over the denominator, and converting an improper fraction to a mixed number by writing the whole number of times the denominator fits into the numerator and the remainder over the denominator.
Rational numbers can be defined as any number that can be made by dividing one integer by another. This includes positive and negative numbers, whole numbers, fractions, and decimals.
To add or subtract fractions, they must first be converted to have a common denominator. This is done by finding the least common multiple of the denominators and using it as the new common denominator.
Multiplying and dividing fractions follows simple rules: for multiplication, multiply the numerators and multiply the denominators; for division, keep the first fraction the same, change the division symbol to multiplication, and flip the second fraction.
The document provides an overview of topics related to numbers and quantities that may appear on the ACT exam, including:
I. Real and complex number systems, rational vs irrational numbers, integers, whole numbers, natural numbers. Converting between fractions, mixed numbers, and improper fractions. Comparing fractions. (Paragraphs 1-13)
II. Counting consecutive integers, using the number line to locate numbers and find distances. Radicals, simplifying square roots. (Paragraphs 14-23)
III. The complex number system, imaginary numbers, and the complex plane. Greatest common factors and least common multiples, using calculators to find them. (Paragraphs 24-25)
1) A mixed number has a whole number part and a fractional part, while an improper fraction has a numerator larger than the denominator.
2) To change between mixed numbers and improper fractions, you can multiply or divide the whole number by the denominator and add or subtract the numerator.
3) When adding, subtracting, multiplying or dividing fractions, you often need a common denominator or need to use reciprocals.
Two's complement representation allows binary arithmetic on signed integers to yield the correct results. Positive numbers are represented as simple binary, while negative numbers are the binary complement of the corresponding positive number. The most significant bit indicates the sign, with 0 being positive and 1 being negative. To calculate the two's complement of a number, invert and add 1 to its binary representation. Two's complement arithmetic follows the same rules as binary arithmetic. Overflow occurs when adding two numbers of the same sign yields a result with the opposite sign.
This document explains fractions including:
- The numerator and denominator define a fraction, with the denominator indicating the number of equal parts in the whole and the numerator indicating how many parts are selected.
- Fractions can be proper, improper, mixed numbers, or equivalent. Equivalent fractions have the same value when simplified to their lowest terms.
- To order fractions, find a common denominator by finding the lowest common multiple of the denominators, then compare the numerators.
This document discusses rational numbers and different types of fractions including mixed numbers, improper fractions, adding, subtracting, multiplying, and dividing fractions. It explains that rational numbers are numbers that can be made by dividing one integer by another. Fractions have a numerator and denominator and can be added or subtracted by finding a common denominator. To multiply fractions, you multiply the numerators and denominators. To divide fractions, you keep the first fraction the same, change the operation to divide, and flip the second fraction to its inverse.
Scientific Measurement can be summarized in 3 sentences:
Measurement involves assigning numerical values to properties of matter using standardized units. The units provide the numerical value and type of quantity being measured, with significant figures indicating the precision. Dimensional analysis allows conversion between different units by canceling unwanted units and introducing desired units using conversion factors derived from unit equality relationships.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
22. Changing a mixed number to an
improper fraction.
1 3
STEP ONE:
Multiply the
DENOMINATOR
and the
WHOLE
NUMBER
8
23. Changing a mixed number to an
improper fraction.
1 3
STEP ONE:
Multiply the
DENOMINATOR
and the
WHOLE
NUMBER
8
24. Changing a mixed number to an
improper fraction.
1 3
STEP ONE:
Multiply the
DENOMINATOR 8x1=8
and the
WHOLE
NUMBER
8
25. Changing a mixed number to an
improper fraction.
1 3
STEP ONE: denominator
Multiply the
DENOMINATOR 8x1=8
and the
WHOLE
NUMBER
8
26. Changing a mixed number to an
improper fraction.
1 3
STEP ONE: denominator
Multiply the
DENOMINATOR 8x1=8
and the
WHOLE
NUMBER
8 whole #
27. Changing a mixed number to an
improper fraction.
1 3
STEP ONE: denominator
Multiply the
DENOMINATOR 8x1=8
and the
WHOLE
NUMBER
8 whole #
answer
(product)
42. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
WHOLE NUMBER
43. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
WHOLE NUMBER
STEP TWO: ADD
the PRODUCT to
the NUMERATOR.
44. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
WHOLE NUMBER
STEP TWO: ADD
the PRODUCT to
the NUMERATOR.
STEP THREE: PLACE answer
OVER original
DENOMINATOR
45. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
M
WHOLE NUMBER
STEP TWO: ADD
the PRODUCT to
the NUMERATOR.
STEP THREE: PLACE answer
OVER original
DENOMINATOR
46. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
M
WHOLE NUMBER
STEP TWO: ADD
A
the PRODUCT to
the NUMERATOR.
STEP THREE: PLACE answer
OVER original
DENOMINATOR
47. Changing a mixed number to an
improper fraction.
STEP ONE: MULTIPLY the
DENOMINATOR and the
M
WHOLE NUMBER
STEP TWO: ADD
A
the PRODUCT to
the NUMERATOR.
P
STEP THREE: PLACE answer O
OVER original
DENOMINATOR D
87. Changing an improper fraction
to a mixed number.
STEP ONE:
DIVIDE the
numerator by the
denominator.
88. Changing an improper fraction
to a mixed number.
STEP ONE:
DIVIDE the
numerator by the
denominator.
STEP TWO:
PLACE REMAINDER
OVER the original
DENOMINATOR
89. Changing an improper fraction
to a mixed number.
STEP ONE: D
DIVIDE the
numerator by the
denominator.
STEP TWO:
PLACE REMAINDER
OVER the original
DENOMINATOR
90. Changing an improper fraction
to a mixed number.
STEP ONE: D
P
DIVIDE the
numerator by the
denominator.
STEP TWO:
R
PLACE REMAINDER
OVER the original
O
DENOMINATOR D