Kentucky K–6 mathematics standards

Count to 100 by ones and by tens.

Count backwards from 30 by ones.

Count forward beginning from a given number within the known sequence within 100 (instead of having to begin at 1).

Write numbers from 0 to 20.

Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

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.

Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Understand that each successive number name refers to a quantity that is one larger.

Count to answer "how many?" questions with as many as 20 things arranged in a line, a rectangular array, or a circle.

Count to answer "how many?" questions with as many as 10 things in a scattered configuration.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group.

Compare two numbers between 1 and 10 presented as written numerals.

Describe objects in the environment using names of shapes.

Describe the relative positions of these objects using terms above, below, in front of, behind and next to.

Correctly name shapes regardless of orientations or overall size.

Identify shapes as two-dimensional or three-dimensional.

Describe the similarities, differences and attributes of two and three dimensional shapes using different sizes and orientations.

Model shapes in the world by building figures from components and drawing shapes.

Compose simple shapes to form larger shapes.

Describe measurable attributes (length, height, weight, width, depth) of an object or a set of objects using appropriate vocabulary.

Directly compare two objects with a measurable attribute in common, to see which object has "more of"/ "less of" the attribute and describe the difference.

Classify and sort objects or people by attributes. Limit objects or people in each category to be less than or equal to 10.

Recognize and identify coins by name (penny, nickel, dime, quarter).

Compose and decompose numbers from 11 to 19 using quantities (numbers with units) of ten ones and some further ones. Understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations.

Solve addition and subtraction word problems and add and subtract within 10 by using objects or drawings to represent the problem.

Decompose numbers into two groups in more than one way by using objects or drawings and record each decomposition by a drawing or equation.

Use objects or drawings to demonstrate equality as the balancing of quantities.

For any number from 1 to 9, find the number that makes 10 when added to the given number by using objects or drawings and record the answer with a drawing or equation.

Fluently add and subtract within 5.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes.

Compose two-dimensional shapes to create rectangles, squares, trapezoids, triangles, half-circles and quarter-circles composite shape and compose new shapes from the composite shapes.

Use 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 shapes.

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. Describe the whole as two of or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

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 same-size length units, by laying multiple copies of a shorter object (the length unit) end to end with no gaps or overlaps.

Tell and write time in hours and half-hours using analog and digital clocks.

Identify the coins by values (penny, nickel, dime, quarter).

Pose a question that can be answered by gathering data.

Determine strategy for gathering data from peers.

Organize and represent data in a table/chart with up to three categories.

Interpret data to answer questions about the table/chart that connects to the question posed, including total number of data points, how many in each category and how many more or less are in one category than in another.

Count forward to and backward from 120, starting at any number less than 120.

In this range, read and write numerals and represent a number of objects with a written numeral.

10 can be thought of as a bundle of ten ones — called a "ten."

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).

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Add within 100 using<ul><li>concrete models or drawings;</li><li>strategies based on place value;</li><li>properties of operations;</li><li>the relationship between addition and subtraction.</li></ul>

Relate the addition strategy to a written method and explain the reasoning used. 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.

Subtract using:<ul><li>concrete models or drawings;</li><li>strategies based on place value;</li><li>properties of operations;</li><li>the relationship between addition and subtraction</li></ul>

Relate the subtraction strategy to a written method and explain the reasoning used.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, by using objects, drawings and equations with a symbol for one unknown number to represent the problem.

Apply properties of operations as strategies to add and subtract.

Understand subtraction as an unknown-addend problem.

Relate counting to addition and subtraction.

Fluently add and subtract within 10.

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making 10; decomposing a number leading to a 10; using the relationship between addition and subtraction; creating equivalent but easier or known sums.

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 three whole numbers.

Recognize and draw shapes having specified attributes, such as a given number of angles or sides. Identify triangles, quadrilaterals, pentagons, hexagons and cubes (identify number of 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. Recognize that equal shares of identical wholes need not have the same shape.

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.

Estimate lengths using units of inches, feet, yards, centimeters and meters.

Measure to determine how much longer one object is than another, expressing the length difference in terms of either a customary or metric standard length unit.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units by using drawings and equations with a symbol for the unknown number to represent the problem.

Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, … and represent whole-number sums and differences within 100 on a number line.

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Solve word problems with adding and subtracting within 100, (not using dollars and cents simultaneously) using the $ and ¢ symbols appropriately (not including decimal notation).

Identify a statistical question focused on measurements.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object.

Show the measurements by making a dot plot, where the horizontal scale is marked off in whole-number units.

Create a pictograph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put together, take-apart and compare problems using information presented in a bar graph.

100 can be thought of as a bundle of ten tens — called a "hundred."

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).

Count forwards and backwards within 1000; skip-count by 5s, 10s and 100s.

Read and write numbers to 1000 using base-ten numerals, number names and expanded form.

Compare two three-digit numbers based on meanings of the hundreds, tens and ones digits, using >, =, and < symbols to record the results of comparisons.

Fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction.

Add up to four two-digit numbers using strategies based on place value and properties of operations.

Represent and solve addition and subtraction problems using…<ul><li>concrete models or drawings;</li><li>strategies based on place value;</li><li>properties of operations;</li><li>the relationship between addition and subtraction and;</li><li>relate drawings and strategies to expressions or equations.</li></ul>

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 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.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, by using drawings and equations with a symbol for the unknown number to represent the problem.

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; write an equation to express an even number as a sum of two equal addends.

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.

Recognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons and hexagons).

Recognize and classify quadrilaterals (rectangles, squares, parallelograms, rhombuses, trapezoids) by side lengths and understanding shapes in different categories may share attributes and the shared attributes can define a larger category.

Identify shapes that do not belong to a given category or subcategory.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Tell and write time to the nearest minute and measure elapsed time intervals in minutes. Solve word problems involving addition and subtraction of time intervals within and across the hour in minutes.

Measure and estimate masses and liquid volumes of objects using standard units of grams (g), kilograms (kg) and liters (L).

Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

Identify a statistical question focused on categorical data and gather data;

Create a scaled pictograph and a scaled bar graph to represent a data set (using technology or by hand);

Make observations from the graph about the question posed, including "how many more" and "how many less" questions.

Identify a statistical question focused on numerical data;

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch.

Show the data by making a dot plot where the horizontal scale is marked off in appropriate units – whole numbers, halves, or quarters.

Make observations from the graph about the question posed, including questions about the shape of the data and compare responses.

Recognize area as an attribute of plane figures and understand concepts of area measurement.

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 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 the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

Recognize area as additive. Find areas of figures that can be decomposed into non-overlapping rectangles by adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Find the perimeter given the side lengths of a polygon.

Find an unknown side length, given the perimeter and some lengths.

Draw rectangles with the same perimeter and different areas or with the same area and different perimeters.

Use place value understanding to round whole numbers to the nearest 10 or 100.

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations and/or the relationship between addition and subtraction.

Multiply one-digit whole numbers by multiples of 10 in the range of 10–90 using strategies based on place value and properties of operations.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Represent a fraction 1/b (unit fraction) on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.<ul><li>Recognize each part has size 1/b.</li><li>a unit fraction, 1/b is located 1/b of a whole unit from 0 on the number line.</li></ul>

Represent a non-unit fraction a/b on a number line by marking off lengths of 1/b (unit fractions) from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the non-unit fraction a/b on the number line.

Understand two fractions as equivalent (equal) if they are the same size, or same point on a number line.

Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent through writing or drawing.

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. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

Interpret and demonstrate products of whole numbers.

Interpret and demonstrate whole-number quotients of whole numbers, where objects are partitioned into equal shares.

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities, by using drawings and equations with a symbol for the unknown number to represent the problem.

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.

Understand division as an unknown-factor problem.

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.

Use various strategies to solve two-step word problems using the four operations (involving only whole numbers with whole number answers). Represent these problems using equations with a letter standing for the unknown quantity. 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.

Draw points, lines, line segments, rays, angles (right, acute, obtuse) and perpendicular and parallel lines. Identify these in two-dimensional figures.

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence of absence of angles of a specified size. Recognize right triangles as a category and identify right triangles.

Recognize a line of symmetry for a two-dimensional figure.

Identify line-symmetric figures and draw lines of symmetry.

Understand the relationship of measurement units within any given measurement system.

Within any given measurement system, express measurements in a larger unit in terms of a smaller unit.

Record measurement equivalents in a two-column table.

Solve measurement problems involving whole number, simple fractions or decimals.

Solve problems that require converting a given measurement from a larger unit to a smaller unit within a common measurement system, such as 2 km = 2,000 m.

Visually display measurement quantities using representations such as number lines that feature a measurement scale.

Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

Identify a statistical question focused on numerical data.

Make a dot plot to display a data set of measurements in fractions of a unit (½, ¼, 1/8).

Solve problems involving addition and subtraction of fractions by using information presented in dot plots.

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint and understand concepts of angle measurement.

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

Recognize angle measure as additive. When an angle is into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.

Recognize 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.

Compare two multi-digit numbers based on meanings of the digit in each place, using >, =, and < symbols to record the results of comparisons.

Use place value understanding to round multi-digit whole numbers to any place.

Fluently add and subtract multi-digit whole numbers using an algorithm.

Multiply whole numbers<ul><li>Up to four digit number by a one-digit number</li><li>Two-digit number by two-digit number</li></ul>Multiply using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

Divide up to four-digit dividends by one-digit divisors. Find whole number quotients and remainders using<ul><li>strategies based on place value</li><li>the properties of operations</li><li>the relationship between multiplication and division</li></ul>Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

Use visual fraction models to recognize and generate equivalent fractions that have different numerators/denominators even though they are the same size.

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b).

Compare two fractions with different numerators and different denominators using the symbols <, =, or >. Recognize comparisons are valid only when the two fractions refer to the same whole. Justify the conclusions.

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Decomposing a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions.

Add and subtract mixed numbers with like denominators.

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

Understand a fraction a/b as a multiple of 1/b.

Understand a multiple of a/b as a multiple of 1/b and use this understanding to multiply a fraction by a whole number.

Solve word problems involving multiplication of a fraction by a whole number.

Convert a fraction with a denominator of 10 to an equivalent fraction with a denominator of 100.

Add two fractions with respective denominators 10 and 100.

Use decimal notation for fractions with denominators 10 or 100.

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.

Record the results of comparisons with the symbols >, =, or < and justify the conclusions.

Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations.

Multiply or divide to solve word problems involving multiplicative comparisons by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Perform operations in the conventional order when there are no parentheses to specify a particular order.

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. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computations and estimation strategies including rounding.

Find all factor pairs for a given whole number.

Recognize that a whole number is a multiple of each of its factors.

Determine whether a given whole number is a multiple of a given one-digit number.

Determine whether a given whole number is prime or composite.

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern not explicit in the rule itself.

Use a pair 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. Understand that the first number indicates how far to travel from the origin in the direction of one axis and the second number indicates how far to travel in the direction of the second.

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.

Classify two-dimensional figures in a hierarchy based on properties.

Convert among different size measurement units (mass, weight, liquid volume, length, time) within one system of units (metric system, U.S. standard system and time).

Identify and gather data for statistical questions focused on both categorical and numerical data. Select an appropriate data display (bar graph, pictograph, dot plot). Make observations from the graph about the questions posed.

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 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. Represent threefold whole-number products as volumes.

Apply the formulas V= l × w × h and V = B × h for rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Recognize volume as additive. 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.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Multiply and divide by powers of 10.<ul><li>Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.</li><li>Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.</li><li>Use whole-number exponents to denote powers of 10.</li></ul>

Read and write decimals to thousandths using base-ten numerals, number names and expanded form.

Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Use place value understanding to round decimals to any place.

Fluently multiply multi-digit whole numbers (not to exceed four-digit by two-digit multiplication) using an algorithm.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors using<ul><li>strategies based on place value</li><li>the properties of operations</li><li>the relationship between multiplication and division</li></ul>

Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

Add, subtract, multiply and divide decimals to hundredths using…<ul><li>concrete models or drawings</li><li>strategies based on place value</li><li>properties of operations</li><li>the relationship between addition and subtraction</li></ul>

Relate the strategy to a written method and explain the reasoning used.

Efficiently add and subtract fractions with unlike denominators (including mixed numbers) by<ul><li>using reasoning strategies, such as counting up on a number line or creating visual fraction models</li><li>finding common denominators</li></ul>

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators.

Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers by using visual fraction models or equations to represent the problem.

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

Find the area of a rectangle with fractional side lengths by tiling it with 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. Multiply fractional side lengths to find areas of rectangles and represent fraction products as rectangular areas.

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.

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Solve real world problems involving multiplication of fractions and mixed numbers.

Interpret division of a unit fraction by a non-zero whole number and compute such quotients.

Interpret division of a whole number by a unit fraction and compute such quotients.

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

Use parentheses, brackets or braces in numerical expressions and evaluate expressions that include symbols.

Write simple expressions with numbers and interpret numerical expressions without evaluating them.

Generate a rule for growing patterns, identifying the relationship between corresponding terms (x, y).

Generate patterns using one or two given rules (x, y).

Use tables, ordered pairs and graphs to represent the relationship between the quantities.

Write and evaluate numerical expressions involving whole-number exponents.

Write expressions that record operations with numbers and with letters standing for numbers.

Identify parts of an expression using mathematical terms (sums, term, product, factor, quotient, coefficient); view one or more parts of an expression in a single entity.

Evaluate expressions for specific values of their variables, including values that are non-negative rational numbers. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Apply the properties of operations to generate equivalent expressions.

Identify when two expressions are equivalent when the two expressions name the same number regardless of which value is substituted into them.

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 substitution to determine whether a given number in a specified set makes an 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.

Write an inequality of the form x > c, x < c, x ≥ c, or x ≤ c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on vertical and horizontal number lines.

Appropriately recognize one quantity as the dependent variable and the other as the independent variable.

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.

Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the question.

Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and quadrilaterals; apply these techniques in the context of solving real-world and mathematical problems.

Find the volume of a right rectangular prism with rational number edge lengths. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with rational number edge lengths in the context of solving real-world and mathematical problems.

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. Apply these techniques in the context of solving real-world and mathematical problems.

Classify three-dimensional figures including cubes, prisms, pyramids, cones and spheres.

Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions.

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.

Fluently add, subtract, multiply and divide multi-digit decimals using an algorithm for each operation.

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.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize 0 is its own opposite and the opposite of a positive number is a negative, and the opposite of a negative number is a positive, such as−(−3)=3.

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.

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize the similarity between whole numbers, their negative opposites and their positions on a number line, ordered pairs differ only by signs and their locations on one or both axes.

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.

Distinguish comparisons of absolute value from statements about order.

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Understand the concept of a unit rate a/b associated with a ratio a:b with B ≠ 0 and use rate language in the context of a ratio relationship.

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 rate problems including those involving unit pricing and constant speed.

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Formulate Questions: Formulate a statistical question as one that anticipates variability and can be answered with data.

Collect Data: Design and use a plan to collect appropriate data to answer a statistical question.

Analyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual and comparing individual to group.

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 numerical 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 to describe a typical value, while a measure of variation describes how the values in the distribution vary.

Display the distribution of numerical data in plots on a number line, including dot plots, histograms and box plots.

Reporting the number of observations.

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

Determining quantitative measures of center (median and/or mean) to describe distribution of numerical data.

Describing distributions of numerical data qualitatively relating to shape (using terms such as cluster, mode(s), gap, symmetric, uniform, skewed-left, skewed-right and the presence of outliers) and quantitatively relating to spread/variability (using terms such as range and interquartile range).

Relating the choice of measures of center and variability to the shape of the data distribution.