Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid"). Understand the relationship between numbers and quantities; connect counting to cardinality. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.
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). Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Order three objects by length; compare the lengths of two objects indirectly by using a third object. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
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.
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). Add up to four two-digit numbers using strategies based on place value and properties of operations. Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.
Explain why addition and subtraction strategies work, using place value and the properties of operations. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ? symbols appropriately. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object.
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories.
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Identify arithmetic patterns (including patterns in the addition table or multiplication table], and explain them using properties of operations.
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Measure and estimate liquid volumes and masses of objects using standard units of grams (g], kilograms (kg], and liters (l).
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a ? b and a ? c.
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
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.
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. 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. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation.
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. 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. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Draw points, lines, line segments, rays, angles (right, acute, obtuse], and perpendicular and parallel lines.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. 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. 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 10. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. A cube with side length 1 unit, called a ""unit cube,"" is said to have ""one cubic unit"" of volume, and can be used to measure volume. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Apply the formulas V = l ? w ? h and V = b ? h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 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 ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 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 100 and the least common multiple of two whole numbers less than or equal to 12.
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Write expressions that record operations with numbers and with letters standing for numbers.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient]; view one or more parts of an expression as a single entity. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism.
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. 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. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 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) wherer is the unit rate. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers.
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number.
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Solve real-world and mathematical problems involving the four operations with rational numbers. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals], using tools strategically. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.


Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions.
Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 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. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. 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. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. 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. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. 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. Know and apply the properties of integer exponents to generate equivalent numerical expressions. Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3= p, where p is a positive rational number. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve real-world and mathematical problems leading to two linear equations in two variables.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Know that straight lines are widely used to model relationships between two quantitative variables. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
DESCRIPTION: These images show 19th and 20th century reconstructions of the world-view of the Greek philosopher Strabo who wrote his famous geography at the beginning of the Christian era and compiled his map from travelersa€™ reports and the a€?writingsa€? of ancients. The contribution of Strabo as a scholar of great stature as philosopher, historian, and geographer, epitomizes the continuing importance of the Greek intellectual heritage - and contemporary practice - to the development of cartography in the early Roman world. Strabo claimed to have traveled widely to bring together an enormous amount of geographical knowledge. In many ways the most interesting passages relating to cartography in Straboa€™s Geographia are those that, although they contain no maps, give an account, for the first time in a surviving text, of how a description of the known world should be compiled. In order to avoid the deformational problems of flat maps, Strabo stated that he preferred to construct his map on a globe large enough to show all the required detail. As with all Greek world maps, the great impediment to study for the historian of cartography is that we have only these verbal descriptions, not the images themselves. It is not clear how we should interpret these familiar graphic similes Strabo employed to describe to his readers the land areas and other features on the world map. Strabo was a lengthy and discursive writer, but demonstrated good critical power in assessing earlier geographical writers and has given us a verbal picture of the known world of the time. The most detailed examination of a term arising from Homeric geography (#105) is in respect of Ethiopians. The function of geography, according to Strabo, is to be an interpreter, not of the whole world, but of the inhabited world.
He goes on to say there is little point in making the meridians converge slightly in such a map, so was it rectangular, a forerunner of something like Mercatora€™s projection?a€?Like Herodotus (#109), Strabo had traveled himself from Armenia and western Italy, from the Black Sea to Egypt and up the Nile to PhilA¦. Strabo has summed up for us the knowledge of the ancient world as it was in the days of the Emperor CA¦sar Augustus of the great Roman Empire, as it was when in far-off Syria the Christ was born and the greater part of the known earth was under the sway of Rome. Strabo begins his book with a detailed account of southern Spain where he tells of her two hundred towns. Large quantities of corn and wine are exported, besides much oil, which is of the first quality, also wax, honey, and pitch . But if anyone disbelieves the evidence of reason, it would make no difference, from the point of view of the geographer, whether we make the inhabited world an island, or merely admit what experience has taught us, namely, that it is possible to sail round the inhabited world on both sides, from the east as well as from the west, with the exception of a few intermediate stretches.
Despite the extension of the geographical horizons of the inhabited world since the time of Eratosthenes, Straboa€™s oikumene [inhabited world] was smaller. He goes on to suggest that the quadrilateral in which the Atlantic lies resembles in shape half the surface of a spinning-wheel, and that the oikumene [inhabited world] resembles a chlamys, a Greek mantle.
He goes on to say there is little point in making the meridians converge slightly in such a map, so was it rectangular, a forerunner of something like Mercatora€™s projection?Like Herodotus (#109), Strabo had traveled himself from Armenia and western Italy, from the Black Sea to Egypt and up the Nile to PhilA¦.
Those best known are situated on the rivers, estuaries, and seas; but the two which have acquired the greatest name and importance are Cordova and Cadiz.
He grows enthusiastic over the richness of this part of southern Spain, famous from ancient days under the name of Tartessus for its wealth.
It appears to have been the grammarian Crates of Mallos (#113), a contemporary of Hipparchus, and a member of the Stoic school of Philosophers, who made the first attempt to construct a terrestrial globe, and that he exhibited the same in Pergamum, not far from the year 150 B.
Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). The number of objects is the same regardless of their arrangement or the order in which they were counted.
For example, “Can you join these two triangles with full sides touching to make a rectangle?
For example, determine the unknown number that makes the equation true in each of the equations 8 + ? In this range, read and write numerals and represent a number of objects with a written numeral. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
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.
Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph. For example, describe a context in which a total number of objects can be expressed as 5 ?7. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ?8. For example, determine the unknown number that makes the equation true in each of the equations 8 ? ?
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. Solve one- and two-step ""how many more"" and ""how many less"" problems using information presented in scaled bar graphs. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Within a single system of measurement, express measurements 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. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Solve problems involving addition and subtraction of fractions by using information presented in line plots. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.
Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. Use operations on fractions for this grade to solve problems involving information presented in line plots.
Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Apply the formulas V = l w h andV = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 1 Computations with rational numbers extend the rules for manipulating fractions to complex fractions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Understand that random sampling tends to produce representative samples and support valid inferences.
Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. 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.
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.
For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
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. 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.
For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. For example, estimate the population of the United States as 3 ? 10 8 and the population of the world as 7 ? 10 9 , and determine that the world population is more than 20 times larger. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1], (2,4) and (3,9], which are not on a straight line. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.


Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 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. The now lost map by Strabo represented the sum total of cartographic knowledge before the Christian era. It is generally accepted, however, that he must have compiled much of this information in the great library at Alexandria, where he had access to many earlier texts now lost. His motives for writing such a geography (so he tells us) were that he felt impelled to describe the inhabited world because of the considerable strides in geographical knowledge that had been made through the numerous campaigns of the Romans and Parthians.
The first stage was to locate the portion of the terrestrial globe that was known to be inhabited. And, as to these stretches, it makes no difference whether they are bounded by sea or by uninhabited land; for the geographer undertakes to describe the known parts of the inhabited world, but he leaves out of consideration the unknown parts of it - just as he does what is outside of it.
Although PythA¦s, Eratosthenes, and perhaps Posidonius had fixed its northern limit on the parallel through Thule [Iceland ?
He recommended that it be at least ten feet (approximately three meters) in diameter and mentions Crates (#113) in this regard. Nevertheless, apart from the reduced size of the inhabited world, the map Strabo envisaged was similar in its overall shape to that drawn by Eratosthenes (#112). He treats Homer (#105) as the first writer on geography, and defends the Homeric picture of the known world as substantially true. Thus he says that the legend of the Golden Fleece brought back from Colchis by the Argonauts reflects the search for gold by early Greeks in areas of the Black Sea. What did Homer mean by saying they were a€?divided in two, some where Hyperion rises and some where he setsa€??
Thus, accepting Eratosthenesa€™ measurement of 252,000 stades for the circumference of the earth, the geographer ought not to include the equatorial zone, since that in Straboa€™s view is uninhabitable.
This suggests that the eastern and western extremities of the oikumene were thought of as tapering and convex. But one needs a large globe, so that the section mentioned, being only a fraction of it, may clearly show the appropriate parts of the oikumene, which win present a recognizable shape to users. But his seventeen volumes -vastly important to his contemporaries - read like a romance to us today, and a glance at the map laid down according to his descriptions is like a vague and distorted caricature of the real thing.
A wall-map had already been designed by order of Augustus (#118) to hang in a public place in Rome - the heart of the Empire - so that the young Romans might realize the size of their inheritance, while a list of the chief places on the roads, which, radiating from Rome, formed a network over the Empire, was inscribed on the Golden Milestone in the Forum.
It is astonishing to think that in the days of Strabo the silver mines employed forty thousand workmen, and produced the modern-day equivalence of approximately $1,800 a day!a€?But we cannot follow Strabo over the world in all his detail.
Some had told him it was a third of the whole habitable world, some that it took four months to walk through the plain only. 21, and a century passed before Pliny wrote An Account of Countries, Nations, Seas, Towns, Havens, Mountains, Rivers, Distances, and Peoples who now Exist or Formerly Existed.
It is astonishing to think that in the days of Strabo the silver mines employed forty thousand workmen, and produced the modern-day equivalence of approximately $1,800 a day!But we cannot follow Strabo over the world in all his detail. Understand for these examples that decomposing into more equal shares creates smaller shares. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. 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. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. All his writings were firmly set in, if not direct extensions of, the work of his predecessors. The world map had to be adjusted to take account of these facts, and thus Strabo almost certainly proceeded by taking Eratosthenesa€™ map - and the criticism of it by Polybius, Crates, Hipparchus, and Posidonius - as the basis for his own work.
Strabo reasoned that it lay in a northern quadrant of a globe, in a quadrilateral bounded by the frigid zone, the equator, and two meridians on the sides. And it will suffice to fill out and complete the outline of what we term a€?the islanda€? by pining with a straight line the extreme points reached on the coasting-voyages made on both sides of the inhabited world. 66A° N], Strabo, like Polybius, refused to believe that human life was possible so far north, and he blamed Pytheas for having misled so many people by his claim that the a€?summer tropica€? becomes the a€?arctic circlea€? at the island of Thule.
On the other hand, if a globe of this size could not be constructed, Strabo was familiar from Eratosthenes with the transformation necessary to draw it on a plane surface. In describing its detailed geography, however, Strabo did not employ, at least overtly, Eratosthenesa€™ division of the world into irregular quadrilaterals or sphragides, but he often used geometric figures or comparisons to everyday objects to describe the general outline of a country.
In some cases, where alternative descriptions are provided, he may have been attempting to collate the outlines of more than one map. But within the Homeric chapters he has a section in which he attempts to analyze navigation of the oceans over the ages. When Homer made Hera say: a€?For I shall see the bounds of fertile earth and Oceanus, father of the godsa€?, what he means, says Strabo, is that the Ocean touches all the extremities of the land.
Instead he should start his analysis with the Cinnamon Country [near the mouth of the Red Sea, Somaliland], about 8,800 stades north of the equator, in the south, and with Ireland in the north. Again, he estimated the length of the oikumene as 70,000 stades and its width as less than 30,000.
He tells us of a people living north of the Tagus, who slept on the ground, fed on acorn-bread, and wore black cloaks by day and night. It seems to have been Cratesa€™ idea that the eartha€™s surface, when represented on a sphere, should appear as divided into four island-like habitable regions. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Understand the connections between proportional relationships, lines, and linear equations. Thus his Historical Memoirs in forty-seven books, now lost, was a continuation of Polybius. For a graticule, Strabo adopted the straight forward rectangular network of parallels and meridians. It is also probable that students were expected to consult the text of the Geographia with the help of maps, so that the shapes thus enumerated may have served as a simple mnemonic. Or again, when Homer describes Odysseus as seeing land as he was on the crest of a great wave, he must have been referring to the curvature of the earth, a phenomenon familiar to sailors.
He categorizes regions from south to north according to greatest length of day in equinoctial hours.
If one cannot make it as big or not much smaller, one should construct a map of the oikumene on a plane surface at least seven feet long.
He does not think Britain is worth conquering - Ireland lies to the north, not west, of Britain; it is a barren land full of cannibals and wrapped in eternal snows - the Pyrenees nun parallel to the Rhine - the Danube rises near the Alps - even Italy herself runs east and west instead of north and south. On the one hemisphere, which is formed by a meridional plane cutting the sphere, lies our own oecumene or habitable world, and that of the Antoecians in corresponding longitude and in opposite latitude; on the other hemisphere lies the oecumene of the Perioecians in our latitude and in opposite longitude, and that of the Antipodes in latitude and longitude opposite to us. The Geographia is of key importance to our whole knowledge of the history of Greek cartography as well as to the history of science in general. The so-called quadrilateral, bounded by half of this arctic circle, half of the equator, and segments of two meridians, is a spherical quadrilateral, a portion of a sphere. He defended his projection on the ground that it would make only a slight difference if the circles on the earth were represented by straight lines, a€?for our imagination can easily transfer to the globular and spherical surface the figure or magnitude seen by the eye on a plane surface.a€? The dimensions of this flat map were also to be generous. Similarly, Strabo compares the shape of Iberia to an ox-hide, the Peloponnese to a plane-leaf; and the northern part of Asia, east of the Caspian, to a kitchen knife with the straight side along the Taurus range and the curved side along the northern coastline. Yet if such suggestions must remain speculative, there can be little doubt that by the early Roman period world maps and globes drawn by Greek scholars were encouraging a distinctively geographical way of thinking about the world. The problems of the armchair geographer are revealed in the journalistic trick of quotes from quotes on an important exploration: a€?Posidonius, says Herodotus, thinks that certain men sent by Neco completed the circumnavigationa€™. Some of this was polemic against Eratosthenes, who would not have if that early epic poetry could contribute anything to scientific theory. Cratesa€™ view was based upon an unorthodox view that the division was north-south rather than the obvious interpretation of east-west. This list, starting at Meroe with thirteen hours and ending at an area north of the Sea of Azov with seventeen hours, is similar to that given by the elder Pliny. A vast number of people dwell along the Guadalquivir, and you may sail up it almost a hundred and twenty miles from the sea to Cordova and the places a little higher. Through the formulation and expression of such a theory the idea of the existence of an antipodal people was put forth as a speculative problem, an idea frequently discussed in the middle ages, and settled only by the actual discovery of antipodal regions and antipodal peoples in the day of great transoceanic discoveries. But he gives us a fuller account of Great Britain, based on the fresh discovers of Roman generals. Armando CortesA?o states that as a source it was a€?second to none in the history of geography and cartographya€? of this period. In this design Strabo had been influenced not only by Eratosthenesa€™ measurement of the earth but also by the concept of the four inhabited worlds, known and unknown, expounded by Crates (#113), to whom he refers explicitly. He estimated the latitudinal extent of the inhabited world as less than 30,000 stades (compared with Eratosthenesa€™ 38,000 stades) and reduced its length to 70,000 stades instead of Eratosthenesa€™ 78,000 stades [one mile = 9 to 10 stades; there has always been some controversy over the equivalent modern length of a stade as used by various authors]. Strabo envisaged that it would be at least seven feet long and presumably three feet wide, which would suit the length of the inhabited world (70,000 by 30,000 stades), one foot being equivalent to 10,000 stades. India, with two adjacent sides (south and east) much longer than the two others, he described as rhomboidal; Mesopotamia, between the Tigris and Euphrates rivers, he saw as being like a boat drawn in profile, with the deck on the Tigris side and the keel near the Euphrates. And it is likely, among the educated group at least, that an increasingly standard image of the inhabited world had come to be more widely accepted through the use of these maps.
This is all he reports, so that we have to beware of using all his work as scientifically worthy material. As mentioned earlier, in the extreme north, Strabo denied the existence of a Thule Island [Iceland?].
A considerable quantity of salted fish is exported, not only from hence, but also from the remainder of the coast beyond the Pillars. That Strabo, at a later date, had this Pergamenian example in mind when stating certain rules to be observed in the construction of globes seems probable, since he makes mention of Cratesa€™ globe. Many of the earlier treatises that touch upon maps are known to us only through Strabo, while the interest of his commentary on these writers is in its critical handling of their theories, albeit he sometimes fails to advance truth by this process. Strabo repeats that the river Nile was described by Eratosthenes as a reversed N, and that its mouth was named after the Greek capital letter delta. Perhaps because he is drawing on an account at second hand, he is afraid to support what may have seemed like science fiction. To him the most northerly inhabited area was Ierne [Ireland], itself a€?only wretchedly inhabitable because of the cold, to such an extent that regions beyond it are regarded as uninhabitablea€?. Let the equator be conceived as a circle on it, and let a second circle be conceived parallel to it, delimiting the frigid zone in the northern hemisphere, and through the poles a circle cutting these at right angles. The eye is also delighted with groves and gardens, which for this district are met with in the highest perfection. Formerly they exported large quantities of garments, but they now send the un-manufactured wool remarkable for its beauty. Strabo alone among ancient writers, so far as we at present know, treats of terrestrial globes, practically such as we find in use at the present day. He does not deal extensively with Hanno the Carthaginian, instead spending much effort on questioning the explorations of Eudoxus of Cyzicus, who must have added to the accumulation of knowledge about the remote parts of Africa. Straboa€™s own view is that there were two groups of Ethiopians, one living in Asia and one in Africa; and that Homer thought likewise, though not to the extent of placing the eastern group in India, of which he had no knowledge. Likewise, if one were to go not more than 4,000 stades [500 Roman miles] north from the center of Britain, one would find an area near Ireland, which like the latter would be barely inhabitable. For fifty miles the river is navigable for ships of considerable size, but for the cities higher up smaller vessels are employed, and thence to Cordova river-boats.
He thought that a globe to be serviceable should be of large size, and his reasoning can readily be understood, for what at that time was really known of the eartha€™s surface was small indeed in comparison with what was unknown. However, this idea of eastern Ethiopians living in some area of India and resembling Indians in appearance and customs persisted throughout antiquity. These are not constructed of planks pined together, but they were formerly made out of a single trunk. There is a superabundance of cattle and a great variety of game, while on the other hand there are certain little hares which burrow in the ground (rabbits).
Should one not make use of a sphere of large dimensions, the habitable regions, in comparison with the eartha€™s entire surface, would occupy but small space. A chain of mountains, rich in metal, runs parallel to the Guadalquivir, approaching the river, sometimes more, sometimes less, toward the north.
Surprisingly, though, it seems not to have been read in Rome in the first century, judging from the fact that it is not even mentioned by the elder Pliny. Whoever would represent the real earth, he says, as near as possible by artificial means, should make a sphere like that of Crates, and upon this draw the quadrilateral within which his chart of geography is to be placed. It is said that formerly the inhabitants of Majorca and Minorca sent a deputation to the Romans requesting that a new land might be given them, as they were quite driven out of their country by these animals, being no longer able to stand against their vast multitudes. For this purpose however a large globe is necessary since the section mentioned, though but a very small portion of the entire sphere, must be capable of containing properly all the regions of the habitable earth and of presenting an accurate view of them to those ho wish to consult it. But it should have a diameter of not less than ten feet; those who cannot obtain a globe of this size, or one nearly as large, had better draw their charts on a plane surface of not less than seven feet. They feed on the fruit, of stunted oak, which grows at the bottom of the sea and produces very large acorns. We can easily imagine how the eye can transfer the figure and extent (of these lines) from a plane surface to he that is spherical. So great is the quantity of fruit, that at the season when they are ripe the whole coast on either side of the Pillars is covered with acorns thrown up by the tides. The meridians of each country on the globe have a tendency to unite in a single point at the poles; nevertheless on the surface of a plane map there would be no advantage if the right lines alone which should represent the meridians were drawn slightly to converge.
The tunny fish become gradually thinner, owing to the failure of their food as they approach the Pillars from the outer sea.



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