# Mathematics Content Standards with Benchmarks Levels 1-4, Grade Levels

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2 M Demonstrate understanding of the concept of subtraction, i.e., as in taking away or separating, from numbers up to twenty. M Demonstrate understanding of the concept of division (i.e., as dividing a set into equal groups, or determining number of groups within a set), including the role of place value. M Demonstrate an understanding of prime numbers and identify prime numbers up to 20. M Multiply and divide with numbers involving decimals, e.g., 2.5 x 3.6 and with pencil and paper and using the calculator. M Subtract whole numbers up to three digits (without borrowing). M Divide whole numbers up to hundreds by one digit. M Add and subtract whole numbers up to four digits using efficient methods and checking answers. M Use proportions to solve one-step real-life problems, i.e., involving percents, dimensions, scales, etc. M Demonstrate an understanding of the times tables for the numbers 1, 2, 5, and 10. M Use rounding and estimation for tens and hundreds. For example, estimate the sum of 406 and 798 (nearest hundred) or estimate the difference between 836 and 425 (nearest ten). M Multiply with two and three digit numbers using efficient written methods including checking answers. M Recognize and use equivalencies between common fractions, decimals, and percents to find part of whole-number quantities, i.e., know common fraction, decimal, and percent equivalents, e.g., 50% = 1/2 =.5, 25% =.25 = 1/4,.75 = 75% = 3/4. M Halve even numbers up to 10 and double whole numbers up to 10. M Demonstrate an understanding that even numbers can be paired and that odd numbers represent amounts that when paired have one remaining. M Identify and calculate equivalent fractions (fourths, thirds, halves, eighths, fifths, and tenths) and simplify fractions to lowest terms. M Compute percents by finding the part, the percent, and the whole. M Identify place value of ones, tens, and hundreds. M Know all pairs of numbers with a total of 10. M Convert improper fractions to mixed numbers and mixed numbers to improper fractions. M Use a calculator to calculate efficiently using whole numbers, fractions, decimals, and percents. M Identify basic functions (+, -, x,, =, on/off) on the calculator and digits (0-9). M Identify multiples of 2, 3, 4, 5, and 10 up to x 10. M Add and subtract fractions (fourths, thirds, halves, eighths, fifths, and tenths) using fractions that include like and unlike denominators and whole and mixed numbers. M Identify fractional parts (1/4, 1/3, 1/2) and whole. M Demonstrate an understanding of the times tables for the numbers 0 to 12. M Multiply and divide by fractions (fourths, thirds, halves, eighths, fifths, and tenths) using fractions that include like and unlike denominators and whole and mixed numbers. M Recognize currency (up to \$20.00) and coins; count and trade pennies, nickels, dimes, and quarters to 100 cents. M Identify factoring of common numbers (e.g., 12 = 4x3 = 2x6 = 2x2x3). M Relate multiplication of fractions and division, i.e., multiplying by 1/4 is equivalent to dividing by 4 and dividing by 1/4 is equivalent to multiplying by 4. M Make and verify change. M Divide numbers by 10 and 100 and back-multiply to check results of division. M Express a relationship between two quantities as a fraction or fractional estimate, e.g., 54 of 352 graduates = 54/352 or about 1/6.

3 M Identify and demonstrate an understanding of fractional parts including 1/8, 1/4, 1/3, 1/2, and whole. M Demonstrate how fractions relate to multiplication and division (e.g., divide these 12 into 3 parts; 1/3 of 12 is 4, 2/3 is 8). M Add and subtract common fractions with like denominators. M Identify improper fractions and mixed numbers. M Identify and write amounts of money using decimals, words, and symbols. M Demonstrate an understanding of decimal notation and place value by reading, writing, ordering, and comparing decimals to two decimal places. M Convert and express simple common fractions as decimals. M Show relationship between decimal system and money, i.e., show that 10 can be written as \$.10, 25 can be written as \$.25, etc. M Identify quantities that are proportional. M Interpret the meaning of ratio and express a relationship between two quantities as a ratio. M Read, write, order, and compare decimals of up to three decimal places. M Identify place value for decimals (tenths, hundredths, and thousandths) and round decimals to one or two places or whole numbers. M Compute percentages when part and whole are given using friendly numbers, e.g., 10%, 25%, 50%, and 75%. M Convert decimals to fractions and percents, fractions to decimals and percents, and percents to fractions and decimals. M Add, subtract, multiply, and divide numbers with decimals. M Read and write large numbers with decimals (e.g., 12.6 million = 12,600,000). M Determine a fraction or percent of a decimal, e.g.,.25 = 1/4 = 25%, 1.5 = 1 1/2 = 150%, etc. M Understand and interpret the meaning of percent, i.e., percent represents a ratio of a part to a whole where the whole is 100. M Read, write, order, and compare simple percentages. M Find given percents of any given number, i.e., what is 5% of 125?, what is 6.5% of 90?, etc.

5 M Interpret temperature from Fahrenheit scale in various situations, including negative temperatures. M Read and record time of day in 24-hour format. M Convert units of time: hours, minutes, and seconds. M Identify customary US units of linear measurement and equivalents: inches, feet, yards, and miles. M Measure length, width, height, and perimeter in inches, feet, and yards using a ruler or tape measure. M Make rough-estimate approximations of standard US measurements. M Read, interpret, and use map legends/keys. M Convert and calculate with linear measurements (inches, feet, yards, miles) and know the relationship of familiar units, e.g., 12 inches in a foot, 3 feet in a yard, 4 cups in a quart and convert units of measure in the same systems. M Use and apply concepts of weight and capacity to solve problems, i.e., know the difference between weight and capacity. M Use, read, compare, and calculate with positive and negative Fahrenheit temperatures, i.e., know that temperature increases as it goes up and decreases as it goes down and that the sign of the temperature changes when crossing the zero degree point. M Calculate times using the appropriate value and convert between time formats (including elapsed time), i.e., know equivalencies for hours, seconds, minutes, days, weeks, months, decades, and centuries. M Directly measure perimeter in linear units and area in square units (sq. in., sq. ft., sq. cm.). M Estimate, measure, and compare weights (pounds, ounces) using simple instruments, graduated in familiar units (ounces and pounds) and know when to use appropriate measures. M Convert and calculate using standard US units of weight: tons, pounds, ounces, etc. M Convert and calculate using standard US units of capacity: ounces, quarts, and gallons. M Demonstrate an understanding of the concept of two-dimensional measurements and square units. M Interpret, calculate, apply rates, and estimate equivalencies involving time such as velocity (mi/hr, ft/sec, m/sec), frequency (calls/hr), consumption (cal/day, kw/hr), flow (gal/min), change (degrees/min, inches/year), and unit rates (cents/min, \$/sq. ft., mi/gal). M Interpret and use scale drawings to solve reallife problems. M Relate the measure of one object to another (e.g., this is about 3 times as long, 6 of these will fit in there) and plan linear spacing in a design (e.g., how many lines of what size can fit on a poster of a certain height?).

6 M Read analog and digital scales on measuring devices including meters, gauges, scales, etc. using various types of units and calibrations. M.3 Geometry: Students will develop and apply concepts of geometric properties, relationships, and methods to explore, analyze, and solve mathematical and real-life problems. M.3.1 M.3.2 M.3.3 M.3.4 Grade Level Grade Level Grade Level Grade Level M Model and use directional and positional vocabulary appropriately. M Demonstrate an understanding of the concepts of sameness and halfness, i.e., identify and show where line(s) of symmetry (i.e., the lines that divide something into 2 equal parts) falls in twodimensional figures. M Recognize, identify, and describe basic geometric shapes (triangle, square, circle, rectangle, hexagon, pentagon, and octagon). M Identify and compare elements of a circle (center, radius, diameter, arc, circumference). M Demonstrate an understanding of perimeter being the measure around the outside edges of squares and rectangles. M Use the four main compass directions (N, S, E, W) for spatial orientation. M Draw two-dimensional (2-D) shapes of specific dimensions. M Calculate circumference of a circle using formulas provided. M Identify and describe the properties of common twodimensional shapes (square, circle, rectangle, triangle) using everyday language (straight, curved, etc.). M Define and correctly use the concept of horizontal and vertical positions. M Use informal visual methods to describe and compare shape, dimension, perimeter, area, angles, and sides in two- dimensional (2-D) and three-dimensional (3-D) objects. M Understand the relationship of angles when you have a system of parallel lines cut by a transversal. M Follow a pattern or model to produce or reproduce a shape or object. M Identify properties, locations, and functions of right angles, i.e., know that a right angle is 90 degrees or a quarter turn, that two right angles make a straight line, and four right angles fill a space. M Show more than one line of symmetry in complex shapes. M Use direction, distance, coordinates, latitude, longitude, simple scales, labels, symbols, and keys to read and use maps and plans. M Interpret concepts of similarity and identify figures that are similar or congruent. M Use graph paper to draw two- dimensional shapes in different orientations on a grid. M Demonstrate understanding of the 360 degree system of measuring angles and rotation. M Calculate the area of squares, rectangles, and triangles and other common figures using given formulas. M Estimate the measure of an angle, accurately measure an angle using a protractor, and draw angles of specific measures using a protractor and ruler.

7 M Recognize, identify, and describe the properties of common three-dimensional shapes, i.e., cube, cylinder, and sphere. M Apply the Pythagorean Theorem using simple numbers and basic right triangles. M Identify triangles based on their properties, i.e., right, isosceles, equilateral, scalene, obtuse, and acute. M Identify common polygons of various shapes, i.e., triangles, quadrilaterals, pentagons, hexagons, and octagons. M Identify parallel, perpendicular, and intersecting lines. M Describe characteristics of angles formed by two intersecting lines, i.e., vertical, supplementary, complementary, adjacent, and corresponding/congruent. M Identify angles of 90 and 45 degrees, right, acute, and obtuse. M Use the secondary directions for spatial orientation (e.g., NW, SW, NE, SE). M Use a map with a coordinate grid. M Create threedimensional objects from twodimensional representations. M.4 Data Analysis and Probability: Students will develop and apply concepts of data analysis and probability to explore, analyze, and solve mathematical and real-life problems. M.4.1 M.4.2 M.4.3 M.4.4 Grade Level Grade Level Grade Level Grade Level M Identify and name various simple visual data (graphs, charts, tables) found in authentic publications. M Solve problems using simple graphs (pictograph, bar, line, and circle), tables, or distances on maps. M Identify, describe, and compare how a change in one variable relates to a change in a second variable, i.e., situations with constant/fixed and varying/variable rates of change. M Develop and draw conclusions from tables and graphs using instructor or student selected information. M Interpret data organized in basic categories and groupings. M Identify, count, extract, and interpret pertinent data organized in lists, tables, and charts. M Demonstrate an understanding of the concept of categories such as shape, size, color, or yes/no responses and know how to count each category for subtotals. M Gather data to answer a posed question and analyze and present data visually.

8 M Collect, label, and order numerical information for a particular purpose (e.g., to count and list stock). M Reorient, reorganize, and reformat simple data, i.e., create a table to record and present numerical information. M Represent information so that it makes sense to others, i.e., demonstrate an understanding that information can be represented in different ways (list, table, or diagram) and the importance of labeling information. M Demonstrate that a table can display the same data as a line or bar graph. M Collect, label, and order numerical information for a particular purpose (e.g., keep a log, etc.). M Demonstrate an understanding that when objects or responses are divided into categories, all data must be included in one and only one category; therefore, categories must identify distinct sets, i.e., find a total from subtotaled categories to verify inclusion of all data. M Find the average (mean), median, mode, and range for a data set. Note: it is important for students to recognize that mean and median numbers are considered averages and that averages represent numbers typical of the data that can support an argument. M Identify and interpret simple graphs, tables, etc. M Demonstrate an understanding of scatter plots, i.e., that each X in a line plot represents one and only one item or response; therefore, it is verifiable that the number of responses is equal to the number of X s. M Identify the minimum, maximum, and spread of a data set and describe the effect of spread on mean and median, i.e., know the minimum or maximum value can greatly affect the mean but will not affect the median. M Read values on and make comparative statements about relative values on a simple bar graph. M Demonstrate an understanding that a graph is a visual representation and that a table arranges information in rows and columns. M Demonstrate an understanding of line graphs, i.e., that lines going up mean increase, lines tilting down mean decrease and that they can vary over time, flat lines mean no change, and use specific vocabulary to describe trends, i.e., sharp increase, plummeted, etc. M Develop an understanding of events as certain, impossible, likely, or unlikely to occur. M Sort graphs and tables by type, i.e., know that a bar graph uses bars of various heights to display amount, line graphs use lines to display changes in amount, and circle or pie graphs represent the whole. M Know when percent figures don t add up to 100% and when numbers and percent figures (50%, 25%, 10%) don t match up, i.e., understand that circle graphs represent 100%. M Determine the probability of simple events, e.g., in the results of tossing a coin or rolling a die, etc. M Demonstrate an understanding that lists and tables can be ordered in different ways such as alphabetically, numerically, or randomly. M Recognize that some visual representations distort actual data (bar widths can provide misleading information) or see where authors of data reports can manipulate data to benefit themselves or malign others in provided materials and know how to recognize who produced a data report and how their interests might affect the report conflict of interest.

9 M Compare relative values on a bar graph, i.e., demonstrate an understanding that comparative statements such as greater than or less than can be made based on the height of the bars and relative numerical terms such as twice or half. M Determine whether or not a graph/table connects to statements made in text using title, data labels, and percent matches, i.e., know how to locate data labels in tables and graphs to verify they match statements. M Support simple statements with data and know statements using double and half or fifty percent are accurate. M Make observations, draw conclusions, compare, and extract information from bar and circle graphs. M Know that probability is the ratio of the potential successful outcomes to total possibilities and state probability as a ratio in multiple forms (colon, words, and fractions) with simple scenarios. M Determine the probability of basic events (e.g., in the results of drawing cards from a deck of cards, chance of baby being born on a certain day of week, etc.) and express the likelihood of an occurrence as a ratio, fraction, or percent. M Reorient, reorganize, restate, summarize, or reformat report data (make graphs) for a particular purpose and audience. M Determine and compare probabilities of chance events (e.g., winning lottery prizes). M Calculate the possible combinations (a selection of items where order doesn t matter) of up to five items in simple, practical situations (e.g., I have 4 tickets and 5 potential guests). M Calculate the possible permutations (an arrangement of items/data in a certain order) of up to five elements in simple, practical situations (e.g., ways to sequence titles of 4 different colors in a pattern).

11 M Use a calculator to make basic calculations and solve simple addition, subtraction, multiplication, and division problems and check solutions. M Solve single step, reallife word problems involving addition, subtraction, multiplication, and division using up to two digit whole numbers. M Determine and use appropriate rounding and estimating techniques. Understand that the number "5" rounds up. M Demonstrate an understanding that a variable represents a missing value in addition and subtraction expressions, e.g., substitute the value for the variable in one-step expressions using whole numbers when the value is given. M Solve simple one-step equations by recognizing that addition and subtraction are inverse operations and that multiplication and division are inverse operations and knowing the unknown of simple equations can be found by using the inverse of the operation present. M Demonstrate an ability to use the symbols > and < in number statements with larger numbers. M Understand and use exponents to represent repeated multiplication, i.e., recognize that exponents indicate the number of times that the base is written as a factor. M Read, write, and compute squares and cubes of whole numbers, i.e., 4(4) = 4 2 = 16 and 2(2)(2) = 2 3 = 8. M Interpret and solve simple (one or two steps) reallife word problems involving addition, subtraction, multiplication, and division. M Identify and apply simple formulas with one or two arithmetical steps for reallife contexts. M Write an equation representing verbal situations with no more than two operations, i.e., translate simple word problems involving unknown quantities into simple equations. M Determine the slope of a line when given two points on the line or the equation of a line and relate it to change. M Write the equation of a simple line when given two points or slope and one point. M Demonstrate an understanding of like terms by combining like terms in simple algebraic expressions. M Demonstrate an understanding of the order of operations and use the order of operations when simplifying algebraic expressions. M Add and subtract integers, i.e., positive and negative numbers. M Multiply and divide integers, i.e., positive and negative numbers. M Calculate square roots of perfect squares, estimate within range of square root value, and demonstrate an understanding of how squaring and taking the square root are related. M Evaluate, add, subtract, multiply, and divide expressions involving exponents.

12 M Develop flexibility in solving problems by selecting strategies, i.e., determine when and how to split a problem into simpler parts to make solving easier. M Compute using the correct order of operations to solve problems including multiplication, division, addition, and subtraction (M, D, A, S). M Apply estimation strategies and mental math to approximate solutions and then use a calculator to calculate solutions to contextual problems containing whole numbers and decimals to two places. M Use the calculator to find squares, square roots, and cubes of whole number quantities, i.e., know the calculator keys that generate squares, square roots, and cubes of numbers. M Demonstrate an understanding of scientific notation, i.e., a shorter way to write large or really small numbers. M Demonstrate an understanding of and solve basic algebraic equations involving multiple steps, e.g., 3x + 25 = 100, 2x 16 = 42, 3y+ 3 = 42, m/5 25 = 200. M Translate word phrases into algebraic expressions and vice versa. M Demonstrate an understanding of substituting values into simple formulas and solving for the unknown value. M Demonstrate understanding of the distributive property, e.g., 75 x 12 = 75 x x 2 and 2(a + 6) = 2a + 12 M Read, write, order, and compare positive and negative numbers and identify positive and negative numbers on a number line. M Solve real-life, multistep word problems involving money, measurement, and other contextual situations using whole numbers, decimals, and percents. For example, solve problems relating to payroll deductions, computing and comparing unit pricing, rebates, discounts, deficits, sales taxes, shipping and handling fees, etc. M Recognize and eliminate extraneous information in word problems.

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