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Minnesota K-12 Academic Standards in Mathematics

Number Relationships: Describe/Interpret and use quantities, relationships between and representations of quantities and number systems. Describe and relate operations. Use strategies and procedures accurately, efficiently and flexibly. Assess the reasonableness of the results.

Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number with one and only one object. Understand that the last number said tells the number of objects counted. Understand that each successive number refers to a quantity that is one more. Name the position of an object in a sequence (ordinal count).

Read, write, compare, order and represent whole numbers from 0 to at least 31 (with 0 representing the count of no objects) to answer the question “how many?” Representations may include numerals, pictures, real objects, picture graphs, spoken words and manipulatives, such as connecting cubes. The numbers from 11 to 19 are composed of a 10 and one, two, three, four, five, six, seven, eight or nine ones.

Count forward, with and without objects, to at least 31. Count backward from 20.

Solve and represent a variety of addition and subtraction contextual situation types using objects, drawings, mental images or equations within 10.

Compose and decompose numbers less than or equal to 10 into pairs in more than one way with objects and pictures. Record each decomposition with a drawing or equation.

Fluently add and subtract within 5.

Count collections of objects up to 120 using groups of 5s or 10s.

Read, write, compare, order and represent whole numbers from 0 to 120. Representations may include numerals, expanded notation, addition and subtraction, pictures, tally marks, number lines and manipulatives such as bundles of sticks, ten frames and base 10 blocks. The numbers 10, 20, 30, 40, 50, 60, 70, 80 and 90 refer to one, two, three, four, five, six, seven, eight or nine groups of 10s.

Count, with or without objects, forward and backward from any given number up to 120.

Use models, pictures or numbers to recognize and describe the place value of numbers between 10 and 120 as a relationship of 𝑛 groups of 10 plus an amount represented by a single digit (𝑛 × 10 + 𝑎).

Estimate amounts up to 120 using benchmarks of 5s and 10s.

Solve contextual situations, up to and including 20, using addition and subtraction strategies of adding to, taking from, part-part-whole, difference between and comparing. Solve for unknowns in contextual situations using objects, drawings and equations with unknowns represented by a symbol in all positions (result, change, start).

Add within 100, including adding a two-digit number with a one-digit number and adding a two-digit number with a multiple of 10 using concrete models, place value language and properties of operations. Understand that in adding two-digit numbers, sometimes it is necessary to compose a new ten.

Decompose numbers less than or equal to 10 into pairs, in more than one way, using objects or drawings. Record each decomposition with a drawing or equation.

Use combinations of 10 to add to the next decade through 100.

Determine the double of any single-digit number.

Represent and solve contextual equal sharing situations where a whole number of items is shared equally among 2 groups. Name the fractional amount using the word “half.”

Count collections of objects using groups of 10s and 100s to 1,000. Represent the counting strategy and the total using words, symbols and pictures.

Read, write, compare, order and represent whole numbers up to 1,000. Representations may include numerals, expanded notation, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives such as bundles of sticks, ten frames and base 10 blocks.

Given a three-digit number, mentally find 10 more or 10 less and 100 more or 100 less than the number. Justify reasoning by referencing a model.

Recognize and describe the place value of numbers between 10 and 1,000 as a relationship of groups of tens, hundreds and thousands plus an amount of a single digit. Know that 100 is 10 tens and a thousand is 10 hundreds or 100 tens.

Use addition and subtraction, within 1,000, to solve contextual situations using concrete models or drawings based on place value, properties of operations and/or the relationship between addition and subtraction. Relate the strategy to a written method.

Use a range of strategies and algorithms based on knowledge of place value and equality to flexibly add and subtract two-digit numbers. Strategies may include decomposition, expanded notation and partial sums and differences. Use place value and properties of operations to explain why strategies work.

Fluently add and subtract, within 20, using mental strategies that include incrementing, compensation or fact families.

Use landmarks of 10 to fluently add to 100 and subtract from 100.

Given a value, mentally find 100 more or 100 less, 1,000 more or 1,000 less and 10,000 more or 10,000 less than the number. Justify reasoning by referencing a model.

Recognize and describe the place value of numbers between 10 and 100,000 as a relationship of groups of ten, hundreds and thousands plus an amount of a single digit. Know that 10,000 is 100 hundreds, 1,000 is 10 hundreds or 100 tens.

Compare and order whole numbers up to 100,000 justifying with place value language, number lines, and other tools using >, = and < symbols to record the results of comparisons.

Estimate sums and differences up to 1,000 using strategies based on benchmarks and place value language.

Use a range of strategies and algorithms based on knowledge of place value and equality to flexibly add and subtract within 1,000. Strategies may include decomposition, expanded notation and partial sums and differences. Explain how the strategies work using place value and the properties of operations.

Represent and solve contextual situations involving multiplication, measurement division and partitive division with single-digit factors using visual models.

Multiply and divide within 144, using strategies such as equal groups, repeated addition, the relationship between multiplication and division or properties of operations. Develop fluency with facts of 2s, 5s, 10s and square products.

Multiply one-digit whole numbers by multiples of 10 and 100 using strategies such as decomposition of factors of ten, place value language, repeated addition and properties of operations.

Partition a whole into halves, thirds, fourths and eighths. Wholes can be circles, rectangles and the distance between 0 and 1 on a number line.

Use pictures and symbols to represent non-unit fractions up to 2 as sums of unit fractions using halves, fourths, thirds and eighths.

Generate equivalent forms of one-half and 1 using fractions with denominators of 2, 4 and 8 and justify why these forms are equivalent using a visual model.

Compare and order unit fractions using visual models and describe how the size of the fraction changes as the denominator changes.

Recognize that in a multi-digit whole number, a digit in one place represents 10 times what it represents in the place to its right.

Compare and order whole numbers from 0 to 1,000,000 with place value understanding, number lines and other tools using >, = and < symbols to record the results of comparisons.

Estimate sums and differences, within 1,000,000, using strategies based on place value, approximation and properties of operations.

Estimate products and quotients of multi-digit whole numbers by using simple multiplicative relationships, approximation and place value to assess the reasonableness of results.

Fluently multiply two numbers from 0 to 12 using flexible strategies based on the associative, commutative and distributive properties of multiplication.

Use place value language to describe how to multiply a number by 10, 100 and 1,000.

Solve contextual situations using division with dividends up to the thousands place and using one-digit divisors. Strategies may include using visual models, partial quotients, the commutative, associative and distributive properties and repeated subtraction.

Solve multi-step contextual situations requiring the use of addition, subtraction and multiplication of multi-digit whole numbers. Use various strategies, including the relationship between operations, the use of technology and the context of the situation to assess the reasonableness of results.

Read, write, represent and plot on a number line fractional values between 0 and 3, including mixed numbers and fractions greater than 1 with denominators of 2, 3, 4, 5, 6, 8, 10 and 12. Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers.

Explain why a fraction 𝑎/𝑏 is equivalent to the product 𝑎 × 1/𝑏 using visual models and language.

Explain why a fraction 𝑎/𝑏 is equivalent to a fraction (𝑛 × 𝑎)/(𝑛 × 𝑏) by using visual models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Compare and order fractions between 0 and 3 and justify reasoning using pictures, position on a number line and selecting, when appropriate, among strategies such as using a common numerator, referencing a benchmark and using a common denominator.

Recognize the relationship between decimals and fractions. Read and write decimals and fractions in both decimal and fraction notations using words, symbols and expanded form. Recall the fraction and decimal equivalent forms of one-half, one-quarter and three-quarters.

Compare and order decimal values to the hundredths and justify using place value language, a number line and models such as dimes, pennies, 10 x 10 grids and base 10 blocks. Use place value concepts to name and model equivalent forms of decimal values.

Use visual models to add and subtract fractions with denominators of 2, 4 and 8 with results up to 2.

Multiply two multi-digit numbers using an efficient strategy. Strategies include decomposing factors into factors, decomposing factors into sums or using an area model. Justify the chosen strategy using properties of operations and place value.

Divide multi-digit numbers by a one-digit or two digit divisor using efficient and generalizable procedures based on knowledge of place value and the properties of operations that may include partial quotients and standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction, a mixed number or a decimal.

Solve multi-step contextual situations requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies including the inverse relationships between operations, the use of technology and the context of the situation to assess the reasonableness of results.

Generate equivalent fractions of the forms 𝑎/𝑏 = (𝑛 × 𝑎)/(𝑛 × 𝑏) and 𝑎/𝑏 = (𝑎 ÷ 𝑛)/(𝑏 ÷ 𝑛) and justify relationships using visual models.

Given a value, mentally find 0.1 more or 0.1 less, 0.01 more or 0.01 less and 0.001 more or 0.001 less than the number. Justify reasoning by referencing a visual model.

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.

Recognize and flexibly generate equivalences between fractions and decimals to the thousandths place and justify using visual models, place value language and symbols.

Compare and order decimal values to the thousandths. Justify using place value language and visual models.

Fluently add and subtract fractions with unlike denominators (including mixed numbers) and justify using equivalent fractions, visual models and the number line.

Solve contextual situations using addition and subtraction of positive rational numbers represented as fractions (including mixed numbers) or decimals using visual models, equations and properties of operations.

Represent multiplication of a whole number of fractional groups, 𝑛 × 𝑎/𝑏, using visual models, including a number line, and explain how the picture shows the product.

Represent contextual multiplication situations of a fractional amount of a whole number amount, 𝑎/𝑏 of a group of 𝑛, using visual models, including a number line, and explain how the picture shows the product.

Represent contextual measurement situations using division of the form 𝑛 ÷ (𝑎/𝑏) where 𝑛 is the total and 𝑎/𝑏 is the amount per group. Use a visual model and explain how the picture shows the number of groups.

Solve multi-step contextual situations using addition and subtraction of positive rational numbers. Use various strategies, including the inverse relationships between operations and the context of the situation, to assess the reasonableness of results.

Locate positive and negative rational numbers on a number line. Plot ordered pairs of positive and negative rational numbers on a coordinate grid.

Interpret statements of inequality (<, >, =) as statements about the relative position of two numbers on a number line, including positive and negative rational numbers in various forms.

Factor whole numbers. Express a whole number as a product of prime factors with exponents. Identify a prime number as a whole number greater than one whose only factors are one and itself.

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 or difference of two whole numbers, from 1 through 100, with a common factor as a multiple of a sum or difference of two whole numbers with no common factor.

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 context.

Estimate solutions to situations with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of the results in the context of the situation.

Multiply and divide fractions and mixed numbers using visual models to represent the situation leading towards generalizable algorithms.

Solve mathematical situations requiring arithmetic, including multiplication and division, with decimals, fractions and mixed numbers, explaining the solution pathway. Interpret quotients in the context of the situation.

Solve situations using the concept of a unit rate 𝑎/𝑏 associated with a ratio 𝑎:𝑏 with 𝑏 ≠ 0 in context, including constant speed and unit pricing to make purchase decisions.

Solve percent situations using visual models including tables of equivalent ratios, tape diagrams or double number lines. Apply concepts of percentage including discounts, markups, tips and commission. Situations can include identifying the part given a whole and the percentage, and identifying the percentage given the part and the whole.

Know that every rational number can be written as the ratio of two integers and as a terminating or repeating decimal. Recognize that 𝜋 is not rational but that it can be approximated by rational numbers such as 22/7 and 3.14.

Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself, such as –(–3) = 3. Recognize that 0 is its own opposite.

Compare positive and negative rational numbers expressed in various forms including the symbols <, >, =, ≤ and ≥.

Recognize subtraction of rational numbers as adding the additive inverse, 𝑝 – 𝑞 = 𝑝 + (–𝑞). Show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in contexts.

Solve mathematical situations involving adding, subtracting, multiplying and dividing positive and negative rational numbers that are integers, fractions and terminating decimals. Use efficient and generalizable procedures. Raise positive rational numbers to whole-number exponents.

Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational.

Use rational approximations of irrational numbers to compare the size of irrational numbers and locate them approximately on a number line. Estimate the value of expressions involving irrational numbers.

Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions.

Multiply and divide numbers expressed in scientific notation. Express answers in scientific notation.

Solve situations in various contexts involving calculating and comparing simple and compound interest. Compound interest situations are limited to compounded annually.

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Number Relationships: Describe/Interpret and use quantities, relationships between and representations of quantities and number systems. Describe and relate operations. Use strategies and procedures accurately, efficiently and flexibly. Assess the reasonableness of the results for Grade 6

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