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Emily Education Resources Education. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs 1, 12 , 2, 24 , 3, 36 , Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Convert among different-sized standard measurement units within a given measurement system e. Count to , starting at any number less than In this range, read and write numerals and represent a number of objects with a written numeral.
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens and 0 ones.
Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e. The numbers , , , , , , , , refer to one, two, three, four, five, six, seven, eight, or nine hundreds and 0 tens and 0 ones. Read and write numbers to using base-ten numerals, number names, and expanded form.
Add up to four two-digit numbers using strategies based on place value and properties of operations. Understand that in adding or subtracting three- digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Mentally add 10 or to a given number —, and mentally subtract 10 or from a given number — Explain why addition and subtraction strategies work, using place value and the properties of operations. Use place value understanding to round whole numbers to the nearest 10 or Multiply one-digit whole numbers by multiples of 10 in the range 10—90 e.
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Fluently add and subtract multi-digit whole numbers using the standard algorithm. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of Use whole-number exponents to denote powers of Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent equal if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e. Explain why the fractions are equivalent, e. Compare two fractions with the same numerator or the same denominator by reasoning about their size.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Use this principle to recognize and generate equivalent fractions. Compare two fractions with different numerators and different denominators, e. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e. Add and subtract mixed numbers with like denominators, e.
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Express a fraction with denominator 10 as an equivalent fraction with denominator , and use this technique to add two fractions with respective denominators 10 and Use decimal notation for fractions with denominators 10 or For example, rewrite 0.
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole.
Commutative property of addition. Associative property of addition. Understand subtraction as an unknown-addend problem. For example, subtract 10 — 8 by finding the number that makes 10 when added to 8. Relate counting to addition and subtraction e. Add and subtract within 20, demonstrating fluency for addition and subtraction within Use strategies such as counting on; making ten e.
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Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. Fluently add and subtract within 20 using mental strategies. Determine whether a group of objects up to 20 has an odd or even number of members, e. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Interpret products of whole numbers, e. Interpret whole-number quotients of whole numbers, e. Use multiplication and division within to solve word problems in situations involving equal groups, arrays, and measurement quantities, e. Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
Apply properties of operations as strategies to multiply and divide. Commutative property of multiplication. Associative property of multiplication.
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Distributive property. Understand division as an unknown-factor problem. Fluently multiply and divide within , using strategies such as the relationship between multiplication and division e. By the end of Grade 3, know from memory all products of two one-digit numbers. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity.
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Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Identify arithmetic patterns including patterns in the addition table or multiplication table , and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Interpret a multiplication equation as a comparison, e. Represent verbal statements of multiplicative comparisons as multiplication equations.
Multiply or divide to solve word problems involving multiplicative comparison, e. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Explain informally why the numbers will continue to alternate in this way. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. Explain informally why this is so. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
Use ratio and rate reasoning to solve real-world and mathematical problems, e. Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Find a percent of a quantity as a rate per e.
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Decide whether two quantities are in a proportional relationship, e. Identify the constant of proportionality unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by equations. Explain what a point x, y on the graph of a proportional relationship means in terms of the situation, with special attention to the points 0, 0 and 1, r where r is the unit rate.
Use proportional relationships to solve multistep ratio and percent problems. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Explain and use the relationship between the sine and cosine of complementary angles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
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Generate multiple samples or simulated samples of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability mean absolute deviation on either team; on a dot plot, the separation between the two distributions of heights is noticeable. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
For example, when rolling a number cube times, predict that a 3 or 6 would be rolled roughly times, but probably not exactly times. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Develop a probability model which may not be uniform by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.
For an event described in everyday language e. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Find the greatest common factor of two whole numbers less than or equal to and the least common multiple of two whole numbers less than or equal to Use the distributive property to express a sum of two whole numbers 1— with a common factor as a multiple of a sum of two whole numbers with no common factor. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values e. Understand a rational number as a point on the number line.