Montana K–6 mathematics standards

Mathematical practice standard 1 is to problem-solve and persevere. Mathematically proficient students: make conjectures, plan, and follow solution strategies; evaluate their progress and accuracy; engage in sense-making and self-monitoring; and persevere in seeking solutions, and value alternative approaches.

Mathematical practice standard 2 is to abstract and generalize. Mathematically proficient students are able to decontextualize and symbolically represent both mathematical and non-mathematical situations to search for and analyze regularities, patterns, and structures.

Mathematical practice standard 3 is to justify and prove. Mathematically proficient students create, evaluate, justify, and refute mathematical claims in developmentally and mathematically appropriate ways.

Mathematical practice standard 4 is to model with mathematics. Mathematically proficient students: make sense of a scenario; identify a problem to be solved, and mathematize it; and apply a mathematical model to reach a solution and verify its viability.

Mathematical practice standard 5 is to represent. Mathematically proficient students recognize, use, create, interpret, and translate representations using appropriate methods and tools; and understand multiple ways of representing mathematical ideas and how they are related.

Mathematical practice standard 6 is to collaborate mathematically. Mathematically proficient students engage in mathematics as a social enterprise through discussion and collaborative inquiry where ideas are offered, debated, connected, and built upon toward solutions, shared understanding, and appreciation of other perspectives.

Mathematical practice standard 7 is to culturally connect. Mathematically proficient students recognize cultural connections and contributions to mathematics, and appreciate the role of mathematics in various cultural contexts, including those of tribally specific Montana Indigenous Peoples.

Flexibly count to 100 by ones and by tens.

Count beginning from a given number within the known sequence.

Write numbers from 0-20 and represent a number of objects with a written numeral 0-20.

Understand the relationship between numbers and quantities and connect counting to cardinality by recognizing that each successive number name refers to a quantity that is one larger within a normal counting sequence.

Count to answer "how many?" in a variety of arrangements and, given a number, produce a set within 20.

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.

Represent addition and subtraction in multiple ways.

Solve addition and subtraction problems in context within 10. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Decompose numbers less than or equal to 10 into pairs in multiple ways.

For any number from 1 to 9, find the number that makes 10 when added to the given number.

Flexibly and accurately add and subtract within 5.

Recognize the characteristics of the commutative property in addition.

Compose and decompose numbers from 11-19 into ten ones and further ones in multiple ways and record each composition or decomposition by a drawing or an equation.

Describe several attributes of a single object.

Directly compare two objects with a measurable attribute in common using comparative language.

Classify, count, and sort objects into categories. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Describe attributes and identify the names of coins.

Explain time in days, months, years, and seasons.

Describe the relative positions of objects in their environment. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Correctly name shapes regardless of their orientations or overall size.

Identify shapes are two-dimensional or three-dimensional.

Analyze and compare two- and three-dimensional shapes using informal language and other attributes.

Model shapes in the environment. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Compose simple shapes to form larger shapes.

Use addition and subtraction within 20 to solve problems of all types. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Solve problems in context that call for addition of three whole numbers with a sum less than or equal to 20 in context of all types. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Flexibly compose and decompose numbers to add and subtract.

Understand subtraction as an unknown-addend problem.

Relate counting to addition and subtraction.

Flexibly, accurately, and efficiently add and subtract within 10.

Use multiple strategies to add and subtract within 20.

Understand the meaning of the equal sign and determine if equations are true or false.

Determine the unknown number in an addition or subtraction equation relating to three numbers.

Flexibly count, read, write, and represent numbers to 120.

Understand that ten is a unit composed of ten ones and that a two-digit number represents tens and ones.

Compare two two-digit numbers using comparison symbols >, =, and <.

Build a foundation for addition within 100 by: adding two-digit to one-digit numbers, and adding multiples of 10 to two-digit numbers.

Using place value, given a two-digit number, find 10 more or 10 less than the number.

Subtract multiples of 10 from a two-digit number.

Order three objects by length and compare the lengths of two objects indirectly by using a third object. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Express the length of an object as a whole number of length units. Understand that the measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

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

Identify the value of coins.

Organize, represent, and interpret data with up to three categories by: asking and answering questions about the total number of data points, identifying how many are in each category, and analyzing differences between categories.

Distinguish between defining attributes versus nondefining attributes.

Build and draw shapes to possess defining attributes.

Compose new shapes using two- and three-dimensional shapes.

Partition circles and rectangles into two and four equal shares. Describe the shares using the words: halves, fourths, and quarters.

Use addition and subtraction within 100 to solve one- and two-step problems in context involving all problem types. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Flexibly, accurately, and efficiently 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.

Use addition to find the total number of objects arranged in rectangular arrays.

Understand one hundred is a unit composed of ten tens and that three-digit numbers represent amounts of hundreds, tens, and ones.

Skip-count by 5s, 10s, and 100s.

Flexibly count, read, write, and represent numbers to 1000.

Compare two three-digit numbers using >, =, and < symbols.

Flexibly, accurately, and efficiently add and subtract within 100 using multiple strategies.

Add up to four two-digit numbers using multiple strategies.

Add and subtract within 1000 using multiple strategies.

Using place value, add or subtract 10 or 100 from a given number.

Understand and make connections between different strategies for addition and subtraction.

Measure the length of an object by selecting and using appropriate tools.

Understand the relationship between unit sizes and number of units by measuring a single object using two different units of common measurement.

Estimate lengths using units of common measurement.

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

Use addition and subtraction within 100 to solve problems in context involving lengths that are given in the same units. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Represent whole numbers as lengths from 0 and represent 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 problems in context involving dollar bills, quarters, dimes, nickels, and pennies using $ and ¢ symbols appropriately.

Generate measurement data and present the data in multiple ways. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Organize, represent, and interpret data with up to four categories. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Solve addition and subtraction problems of all types using data presented.

Recognize and draw shapes having specified attributes.

Partition a rectangle into rows and columns of same-size squares and find the total number.

Partition circles and rectangles into equal shares, recognize that equal shares need not have the same shape, and express the shares in two-halves, three-thirds, and four-fourths.

Understand products of whole numbers as the total number found by multiplying a number of groups by the number of objects per group.

Understand whole-number quotients of whole numbers: as the number of objects in each group with the total quantity divided equally into a number of shares, and as the number of shares when a total number of objects is partitioned into equal-sized groups.

Use multiplication and division within 100 to solve problems in context in situations involving equal groups, arrays, and measurement quantities.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Apply the commutative property of multiplication, associative property of multiplication, and distributive property of multiplication over addition on whole numbers as strategies to multiply.

Use division as an unknown factor problem.

Flexibly, accurately, and efficiently multiply and divide within 100, using strategies such as the relationship between multiplication and division.

Solve two-step problems in context using the four operations, represent these problems using equations with a letter standing for the unknown quantity and assess the reasonableness of answers using mental computation and estimation strategies. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Identify arithmetic patterns and explain them using properties of operations.

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

Flexibly, accurately, and efficiently 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 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 and understand a fraction a/b as the quantity formed by a parts of size 1/b.

Understand a fraction as a number on the number line by: representing a unit fraction on a number line, representing a fraction as multiple copies of a unit fraction on a number line, and representing fractions on a number line.

Understand the equivalence of fractions in special cases and compare fractions by reasoning about their size by: understanding two fractions as equivalent if they are the same size or the same point on a number line, recognizing and generating simple equivalent fractions and by demonstrating or justifying why the fractions are equivalent, writing whole numbers as fractions, recognizing fractions that are equivalent to whole numbers, and locating them on a number line, comparing two fractions with the same numerator or the same denominator by reasoning about their size and recognizing that comparisons are valid only when the two fractions refer to the same whole, recording the results of fraction comparisons with the symbols >, =, and or < and justifying the conclusions.

Tell and write time on an analog and digital clock to the nearest minute and measure time intervals in minutes and solve word problems in context involving addition and subtraction of time intervals in minutes.

Measure and estimate liquid volumes and masses of objects using customary and metric units by adding, subtracting, multiplying, and dividing to solve one-step problems in context that involve masses or volumes that are given in the same units.

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch and show the data by making a line plot where the horizontal scale is marked off in appropriate units.

Recognize area as an attribute of plane figures and understand concepts of area measurement by: understanding that 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, and understanding that 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.

Relate area to the operations of multiplication and addition by: finding the area of a rectangle with whole-number side lengths by tiling it, and showing that the area is the same as would be found by multiplying the side lengths, multiplying side lengths to find areas of rectangles with whole-number side lengths while solving problems in context and representing whole-number products as rectangular areas, using tiling and area models to represent the distributive property in finding area of a rectangle with whole-number side lengths a and b+c is the sum of a x b and a x c, recognizing area as additive, and finding areas of straight line figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts.

Solve problems in context 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.

Understand that shapes in different categories may share attributes and that the shared attributes can define a larger category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

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

Interpret a multiplication equation as a multiplicative comparison, and represent verbal statements of multiplicative comparisons as multiplication equations.

Multiply or divide to solve problems in context that involve multiplicative comparison and distinguish multiplicative comparison from additive comparison.

Solve multistep problems in context 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, and assess the reasonableness of answers using mental computation and estimation strategies. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Find all factor pairs for a whole number in the range 1-100, recognize that a whole number is a multiple of each of its factors, determine whether a given whole number in the range 1-1000 is a multiple of a given one-digit number, and determine whether a given whole number in the range 1-100 is prime or composite.

Analyze a number or shape pattern that follows a given rule, identify and explain informally features of the pattern that were not explicit in the rule itself, generate terms in the resulting sequence, and observe the pattern.

Recognize that in a multi-digit whole number, each place represents ten times the place to its right.

Read and write multi-digit whole numbers using standard form, word form, and expanded form and compare two multi-digit numbers based on the value of the digits in each place using >, =, and < symbols.

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

Accurately and efficiently add and subtract multi-digit whole numbers using the standard algorithm.

Multiply a whole number of up to four digits by a one-digit whole number, multiply two two-digit numbers, flexibly using strategies based on place value and the properties of operations, and illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models and use this principle to recognize and generate equivalent fractions.

Compare two fractions with different numerators and different denominators by creating common denominators or numerators, or by comparing to a benchmark fraction, recognize that comparisons are valid only when the two fractions refer to the same whole, record the results of comparisons with symbols >, =, or <, and justify the conclusions.

Understand a fraction a/b with a > 1 as a sum of fractions 1/b by: understanding 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, adding and subtracting mixed numbers with like denominators by replacing each mixed number with an equivalent improper fraction or other efficient strategies, and solving problems in context that involve addition and subtraction of fractions referring to the same whole or having like denominators. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number by: understanding a fraction a/b as a multiple of 1/b and recording the conclusion by the equation a/b = a*(1/b), understanding a multiple of a/b as a multiple of 1/b, using this to multiply a fraction by a whole number and recognizing n × (a/b) = (n × a)/b), and solving problems in context involving multiplication of a fraction by a whole number. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

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 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.

Know relative sizes of units within one system of measurement and within the system, express measurements of a larger unit in terms of a smaller unit.

Use the four operations to solve problems in context using distances, intervals of time, liquid volumes, masses of objects, and money, including problems with simple fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit, and represent measurement quantities using diagrams that feature a measurement scale. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Apply the area and perimeter formulas for rectangles including problems in context.

Make a line plot to display a data set of measurements in fractions of a unit and solve problems involving addition and subtraction of fractions by using information presented in line plots.

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint and understand concepts of angle measurement by: understanding that an angle is formed by two rays with a common endpoint at the center of a circle that measures a total of 360 degrees, and a single-degree unit measure is equal to 1/360th of the circle, and, and understanding that an angle that turns through n one-degree angles is said to have an angle measure of n degrees.

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

Recognize angle measure as additive. When an angle is decomposed into nonoverlapping 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 including problems in context.

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

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 right triangles as a category, and identify right triangles.

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, identify line-symmetric figures, and draw lines of symmetry. This standard should incorporate designs and cultural context relating to Montana Indigenous Peoples and local communities.

Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols using the order of operations.

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

Generate two numerical patterns using given rules and complete an input-output table for the data, identify apparent relationships between corresponding terms, form ordered pairs from the values in the input-output table, and graph them on a coordinate plane.

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.

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10, and use whole-number exponents to denote powers of 10.

Read, write, and compare decimals to thousandths by: reading and writing decimals to thousandths using standard form, word form, and expanded form, comparing two decimals to thousandths based on meanings of the digits in each place using >, =, and and < symbols.

Use place value understandings to round decimals to any place.

Accurately and efficiently multiply multi-digit whole numbers using the standard algorithm.

Flexibly, accurately, and efficiently find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division and illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawings. This standard should incorporate designs and cultural context relating to Montana Indigenous Peoples and local communities.

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions.

Solve problems in context that involve addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, and 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) and solve problems in context that involve division of whole numbers leading to answers in the form of fractions or mixed numbers.

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction by: expressing 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, finding the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, showing that the area is the same as would be found by multiplying the side lengths, multiplying fractional side lengths to find areas of rectangles, and and represent representing fraction products as rectangular areas.

Interpret multiplication as scaling (resizing), by: 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 explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number, and and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Solve problems in context that involve multiplication of fractions and mixed numbers. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions by: expressing division of a unit fraction by a nonzero whole number and computing such quotients, expressing division of a whole number by a unit fraction and computing such quotients, and solve problems in context involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step problems in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Make a line plot to display a data set of measurements in fractions of a unit and use operations on fractions to solve problems involving information presented in line plots.

Recognize volume as an attribute of solid figures and understand concepts of volume measurement by: understanding that 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, and understanding that 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 non-standard units.

Relate volume to the operations of multiplication and addition and volume problems including problems in context by: finding the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes and showing 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 and representing the product of three whole numbers using the associative property of multiplication, applying 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 including problems in context, recognizing volume as additive and finding volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, and applying this technique to solve problems in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines at 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 x-coordinate, the first number, indicates how far to travel from the origin in the direction of the x-axis and the y-coordinate, the second number, indicates how far to travel in the direction of the y-axis.

Represent problems including problems in context by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation. This standard should incorporate designs and cultural context relating to Montana Indigenous Peoples and local communities.

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.

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.

Use ratio and rate reasoning to solve proportional problems in context about unit rates, percentages (as a rate per 100), and/or measurement units using tables or equations. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Represent, interpret, and compute quotients of fractions and solve problems in context involving division of fractions by fractions.

Accurately and efficiently divide multi-digit numbers using the standard algorithm.

Accurately and efficiently 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. 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 and use positive and negative numbers to represent quantities in problems in contexts, explaining the meaning of 0 in each situation.

Understand a rational number as a point on the number line and extend number line diagrams and coordinate axes by: recognizing opposite signs of numbers as indicating locations on opposite sides of 0 on the number line, recognizing that the opposite of the opposite of a number is the number itself, and that 0 is its own opposite, understanding signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane and recognizing that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes, and, and finding and positioning integers and other rational numbers on a horizontal or vertical number line diagram and finding and positioning pairs of integers and other rational numbers on a coordinate plane.

Understand ordering and absolute value of rational numbers by: interpreting statements of inequality as statements about the relative position of two numbers on a number line diagram, writing, interpreting, and explaining statements of order for rational numbers in problems in context, understanding the absolute value of a rational number as its distance from 0 on the number line and interpreting absolute value as magnitude for a positive or negative quantity in problems in context, and distinguishing comparisons of absolute value from statements about order.

Graph points in all four quadrants of the coordinate plane and include the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Write and evaluate numerical expressions involving whole-number bases and exponents.

Write, read, and evaluate expressions with variables by: writing expressions that record operations with numbers and with variables, identifying parts of an expression using mathematical terms (sum, product, difference, quotient, term, factor, coefficient, variable) and writing expressions that represent verbal descriptions of problems in context, evaluating expressions at specific values of their variables including expressions that arise from formulas performing arithmetic operations, and including those involving whole-number exponents and using the order of operations.

Apply the properties of operations including the distributive property, to generate equivalent expressions and determine when two expressions are equivalent.

Understand how to solve an equation or inequality as a process by using substitution to determine whether a given number in a specified set makes an equation or inequality true.

Write expressions when solving problems in context and understand that a variable can represent an unknown number, or any number in a specified set.

Solve problems including problems in context 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 or x < c to represent a constraint or condition in problems including problems in context; graph and describe solutions of such inequalities on number line diagrams.

Use variables to represent two quantities that change in relationship to one another analyze the relationship between the dependent and independent variables using graphs and tables, and write an equation to express one quantity in terms of the other. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Find the area of triangles, quadrilaterals, and other polygons by composing them into rectangles or decomposing them into triangles and other shapes. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Find the volume of a right rectangular prism with fractional edge lengths by filling it with unit cubes of the appropriate unit fraction edge lengths and connect and apply the formulas V = l*w*h and V = B*h to find volumes of right rectangular prisms with fractional edge lengths to solve problems in context.

Draw polygons in the coordinate plane given coordinates for the vertices, find the length of a horizontal or vertical side, and apply these techniques to problems in context.

Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures in problems including problems in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

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 that can be described by its center, spread, and overall shape. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Recognize that measures of central tendency 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, and describe any overall pattern and any striking deviations from the overall pattern with reference to the context which the data were gathered.

Characterize numerical data sets from a sample in relation to their context, such as by:: reporting the number of observations, describing the nature of the attribute under investigation, including how it was measured and its units of measurement, finding quantitative measures of central tendency (mode, median, and/or mean) and variability (interquartile range and/or mean absolute deviation), and for numerical data sets and relating the choice of measures of central tendency and variability to the shape of the data distribution and the context in which the data were gathered.