Kindergarten
Counting and Cardinality | |
Concepts | • Can connect counting to cardinality (the last number said is the number of things you have)• Understands the relationship between and quantities• Can compare quantities |
Skills | • Knows how to count by 1s, 2s, 5s, and 10s (starting from zero)• Has 1 to 1 correspondence• Knows written number symbols to 30 |
Place Value in Base 10 | |
Concepts | • Can understand numbers 1 to 20 as tens and ones |
Skills | • Can show numbers 1 to 20 as tens and ones using manipulatives or pictures |
Addition | |
Concepts | • Addition is knowing more and less• 1 to 1 correspondence |
Skills | • Writing the numbers 1-10• Counting the groups being added• Composing a number sentence
• Understanding symbols +, = • Can combine numbers up to 10 using manipulatives or other strategies |
Subtraction | |
Concepts | • Subtraction is knowing more and less.• number recognition• 1 to 1 correspondence
• decomposing groups of numbers/objects to the number 10 |
Skills | • write the numbers 1-10• can remove objects to find the total left• understanding of symbols – and =
• can compose a number sentence • can decompose numbers from 10 using manipulatives or other strategies |
Fractions | |
Concepts | • Understand the meaning of half, quarter, and that fractions are equal parts of a larger object |
Skills | • Can identify sections into 1/2, 1/4 |
Algebraic Thinking | |
Concepts | • Understand addition as putting together and adding to, and understand subtraction is taking apart and taking from• Understands that an equal sign represents equality |
Skills | • Show addition and subtraction with manipulatives, mental math, clapping, drawing, verbal explanations, equations• Can identify mathematical patterns and extend them |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • Understand that objects can be measured by length, height, weight, circumference• Understand that time is a measurement• Understand that different money has different value |
Skills | • Use standard/non-standard units in measuring• Can compare and describe measurement results with on another• Tell time to the hour and half hour
• Identify name and value of coins |
Data | |
Concepts | • Understand the likelihood (or chances) of a situation• Understand that things can be classified by category according to their attributes and represented in a graph |
Skills | • Classify and quantify and represent known information into given categories• Gather/use specific information to represent in a graph• count, compare, draw conclusions about how much in each category
• read/interpret bar graphs, venn diagrams and tallies |
Geometry | |
Concepts | • Understand that there are different shapes and to be aware that shapes have names• Shapes are grouped by attributes• Understands that shapes can be 2-D and 3-D |
Skills | • Can recognize and draw shapes having specified attributes.• Can correctly recognize shapes regardless of their orientation and size• Can discover and identify shapes in different environments
• Can group and sort shapes by their attributes |
Grade 1
Numbers in Base 10 | |
Concepts | • Understands that the first digit in a 2-digit number is the number of tens and the next is the number of ones• Understands the first digit in a 3-digit number represents the hundreds place. |
Skills | • Can read, write, and count to 120, starting at any number• Can decompose a two-digit number into tens and ones multiple ways (2 tens + 7 ones or 1 ten + 17 ones• Can compare 2-digit numbers using >, <, =
• Can count by 2s, 5s, and 10s starting on any multiple of 2, 5, or 10 |
Addition | |
Concepts | • Understanding of base 10 system• order in addition doesn’t matter• numbers are made of tens and ones |
Skills | • Relate counting to addition• Using mental math• Numbers can be split into 100s, 10s and 1s
• Can combine numbers up to 20 using manipulatives and other strategies • Can recall facts 1-10 either from memory or using mental strategies such as doubles +1 • understanding a word problem and pulling relevant information to solve • relating a number sentence to a word problem |
Subtraction | ||
Concepts | • can decompose numbers up to 20 using manipulatives or other strategies• can understand why and when to use subtractions• understands that order does matter in subtractions | |
Skills | • relates counting to subtraction• can understand that subtraction is an unknown addend problem• can recall facts 1-10 either from memory or using mental strategies such as doubles -1 | |
Fractions | ||
Concepts | • Understand that one whole unit can be divided equally into parts such as 1/2, 1/4• Understand that the equal parts can be brought back together to make one whole unit• Understand what a fraction is | |
Skills | • Can divide the whole equally into 1/2 and 1/4• Can label parts of things 1/2 and 1/4• Can explain what a fraction is | |
Algebraic Thinking | |
Concepts | • Understand and apply properties of operations and the relationship between addition and subtraction• Understand estimation is an important strategy when adding and subtracting |
Skills | • Can correctly use symbols >, <, and = to compare numbers• Can identify and extend the rule in growth patterns |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • Understand that objects can be measured by length, height, weight, circumference• Understand that time is a measurement• Understand that different money has different value |
Skills | • Use standard/non-standard units in measuring• Can compare and describe measurement results with on another• Tell time to the hour and half hour
• Identify name and value of coins |
Data | |
Concepts | • Understand the likelihood (or chances) of a situation• Understand that things can be classified by category according to their attributes and represented in a graph |
Skills | • Classify and quantify and represent known information into given categories• Gather/use specific information to represent in a graph• count, compare, draw conclusions about how much in each category
• read/interpret bar graphs, venn diagrams and tallies |
Geometry | |
Concepts | • Understand that there are different shapes and to be aware that shapes have names• Shapes are grouped by attributes• Understands that shapes can be 2-D and 3-D |
Skills | • Can recognize and draw shapes having specified attributes.• Can correctly recognize shapes regardless of their orientation and size• Can discover and identify shapes in different environments
• Can group and sort shapes by their attributes |
Grade 2
Numbers in Base 10 | |
Concepts | • Understand that the three digits of a 3-digit number are hundreds, tens, and ones. |
Skills | • Can read, write, and count to 1,000 starting at any number• Can decompose a 3-digit number into 100s, 10s, and 1s in multiple ways• Can compare 3-digit numbers using <, >, and =
• Can count by 2s, 5s, 10s, and 100s starting on any multiple of 2, 5, 10, or 100 |
Addition | |
Concepts | • Numbers can be decomposed and recomposed• Subtraction is the opposite of addition |
Skills | • Can add 2-digit numbers efficiently and correctly using some strategies• Can recall facts 1-20 either from memory or using mental strategies such as double +1• Can correctly determine when to use addition to solve a story problem
• Can use subtraction to solve a missing addend problem and to check their addition solutions • can estimate to assess reasonableness of answer when adding whole numbers |
Subtraction | |
Concepts | • numbers can be decomposed from a large number into smaller numbers• subtraction is the opposite of addition |
Skills | • can subtract 2 digit numbers using some strategy• can recall facts 1-20• to know when to use subtraction to solve a problem
• can estimate to assess reasonableness of answer |
Multiplication | |
Concepts | • Multiplication is a faster way to add equally sized groups |
Skills | • Can skip count using a 100s chart |
Fractions | |
Concepts | • Understand that in order for something to be split into halves, thirds, or fourths, the pieces need to be equal• Understand that 2 halves of something do not have to be identical in shape, just size |
Skills | • Can identify halves, thirds, and fourths, and can label pieces and portions as such• Can divide shapes into halves, thirds, and fourths |
Algebraic Thinking | |
Concepts | • Understands what makes a number even or odd |
Skills | • Can correctly use the symbols >, <, and = to compare numbers and equations• Can create a growth pattern and define the rule of the pattern• Can fill in a missing number in a simple addition and subtraction number sentence. |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • There are different systems of measurement• Different tools are used for different purposes |
Skills | • Be able to make reasonable estimates and then measure standard length, weight, time, and volumes by selecting and using appropriate tools• Can measure to determine how much longer, shorter, etc. something is (can measure and make comparisons• Can tell and write time from analog and digital clocks to the nearest 5 minutes
• Can determine the area and perimeter of a shape using various tools and strategies |
Data | |
Concepts | • Understand the likelihood (or chances) of a situation• Understand that things can be classified by category according to their attributes and represented in a graph |
Skills | • Classify and quantify and represent known information into given categories• Gather/use specific information to represent in a graph• count, compare, draw conclusions about how much in each category
• read/interpret bar graphs, venn diagrams and tallies |
Geometry | |
Concepts | • Understand that there are different shapes and to be aware that shapes have names• Shapes are grouped by attributes• Understands that shapes can be 2-D and 3-D |
Skills | • Can recognize and draw shapes having specified attributes.• Can correctly recognize shapes regardless of their orientation and size• Can discover and identify shapes in different environments
• Can group and sort shapes by their attributes |
Grade 3
Numbers in Base 10 | |
Concepts | • Understand that the four digits in a 4-digit number are thousands, hundreds, tens, and ones. |
Skills | • Can round to the nearest 10 or 100.• Can use place value to find 1,000, 100, or 10 more or less than a number• Can count by 2s, 5s, 10s, and 100s starting on any number
• Can read, write, and count to 10,000 starting on any number • Can read, write, and count to 10,000 starting on any number • Can decompose numbers to 10,000 |
Addition | |
Concepts | • Numbers can be composed and decomposed• Addition is the opposite of subtraction• Numbers can be composed into tens, hundreds, and thousands |
Skills | • Can compose and decompose numbers to 1,000,000• Use multiple strategies to add• Ability to justify/defend procedures and answers
• Recall basic facts readily (1-20) • Can use the standard algorithm to add multi-digit whole numbers |
Subtraction | |
Concepts | • numbers can be composed and decomposed• subtraction is the opposite of addition• numbers can be composed into tens, hundreds, and thousands
• order matters |
Skills | • can compose and decompose numbers to 1,000,000• use multiple strategies• recall basic facts readily (up to 20)
• can use standard algorithm (and regroup as necessary) to subtract multi-digit whole numbers |
Multiplication | |
Concepts | • Multiplication is repeated addition• Multiplication and division are inversely related, as shown through visual models |
Skills | • Can figure out the basic facts up to 12s• Can multiply a double digit by a single digit number using multiple strategies• Can multiply any number by 10, 100, 1000 |
Division | |
Concepts | • Multiplication and division are inversely related• Division is repeated subtraction• Division is making groups and sharing |
Skills | • Can divide numbers within 100, expressing remainders with an understanding of their meaning• Can interpret and solve story problems requiring division• Can read and express equations using different division symbols |
Fractions | |
Concepts | • Understands that fractions are equal parts of a whole• Understands that a whole can be different sized objects or multiple objects• Understands fractions as a number on the number line
• Understands simple equivalencies (1/2 = 2/4) |
Skills | • Can determine fractional parts of a whole• Can represent benchmark fractions on a number line 1/4, 1/2, 1/3, 1/10• Can generate and explain simple equivalencies and compare fractions
• Can express whole numbers as fractions • Can compare two fractions with the same numerator or the same denominator by reasoning about their size |
Algebraic Thinking | |
Concepts | • Numbers can be represented in different ways• Numbers create patterns. These patterns can be used to solve mathematical problems. |
Skills | • Can solve two-step word problems using 4 operations• Can represent problems using equations that include variables to represent unknown quantities• Can identify and explain patterns |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • There are different systems of measurement• Different tools are used for different purposes |
Skills | • Be able to make reasonable estimates and then measure standard length, weight, time, and volume to fractions of units by selecting and using appropriate tools• Can measure to determine how much longer, shorter, etc. something is (can measure and make comparisons)• Can tell and write time from analog and digital clocks to the nearest minute
• Can solve problems involving elapsed time • Can determine area and perimeter of a shape using various tools and strategies |
Data | |
Concepts | • Information can be collected and organized (data)• Understand that there are factors that influence the likelihood of certain outcomes (probability)• Understand the real-world applications involved in data and probability |
Skills | • Be able to create and interpret a visual representation of a data set using an appropriate method.• Conduct a probability simulation and interpret the outcome |
Geometry | |
Concepts | • Shapes can be classified by their attributes• Understands the difference between 2-D and 3-D images• Understands shapes as a combination of angles and lines
• Shapes within categories can vary • Understands symmetry • Understands categories of lines and their properties • Understands coordinate system |
Skills | • Can sort shapes by attribute• Can identify different shapes that have the same attributes• Can find lines of symmetry for 2-D figures
• Can measure angles using a protractor and/or formulas — recognize it as additive • Can graph points on the coordinate plane to represent real world and mathematical problems |
Grade 4
Numbers in Base 10 | |
Concepts | • Understand that the five digits of a 5 digit number are ten-thousands, thousands, hundreds, tens, and ones.• Understands that numbers can be written in expanded and standard form. |
Skills | • Can round to the nearest thousand or ten thousand• Can read, write, and count to 100,000• Can decompose numbers to 100,000 by place value in multiple ways |
Addition | |
Concepts | • Addition is related to multiplication• You can add parts of whole numbers (fractions) as well as whole numbers |
Skills | • Fluently add multi-digit whole numbers using the standard algorithm• Ability to solve multi-step word problems involving addition• Can use multiple strategies and defend solutions
• can add fractions with like denominators |
Subtraction | |
Concepts | • subtraction is related to division• you can subtract parts of whole numbers as well as whole numbers |
Skills | • fluently subtract multi-digit numbers using standard algorithm with regrouping• ability to solve multi-step word problems involving subtraction• can subtract fractions with like denominators |
Multiplication | |
Concepts | • The relationship between multiplication and addition can be used as a strategy• Understand relationship to division as an opposite operation and use strategies based on properties of operations |
Skills | • Estimate and compare products using >, < and = symbols• Ability to multiply up to 4 digits by 1 or 2 digits with regrouping• Know multiplication facts fluently to the 12s
• Can multiply fluently by 10s, 100s, 1000s • Illustrate and explain calculations using equations, arrays, and/or models • Can solve story problems involving multiplication. |
Division | |
Concepts | • Understand the relationship between multiplication and division• Understand divisibility rules and patterns• Understand that division involves both sharing and grouping |
Skills | • Can identify factors and products• Can identify divisors, dividends, and quotients• Can perform simple long division with 1 digit divisors with remainders
• Can use and read a variety of division algorithms • Knows division facts up to 144 (12s) • Can interpret needed information in story problems when it involves division, to include remainders with accuracy • Can check quotients by engaging in reverse multiplication operation — adding remainders as needed |
Fractions | |
Concepts | • Understand that fractions represent parts of a whole• Understand addition and subtraction of fractions and separating parts of the same whole• Understand concepts of mixed numbers and improper fractions
• Understand equivalencies |
Skills | • Can decompose a fraction into a sum of fractions with the same denominator• Add and subtract mixed numbers with like denominators• Can determine equivalencies among fractions
• Can use models to represent fractional understanding • Solves story problems involving the addition and subtraction of fractions referring to the same whole |
Decimals | |
Concepts | • Understands that fractions can be represented as decimals and vice versa• Understands that the first two places to the right of the decimal point represent tenths and hundredths |
Skills | • Can read, write, and compare decimals to the hundredths place• Can express a fraction with a denominator with ten or one hundred as a decimal and vice versa• Can add and subtract decimals to the hundredths place |
Algebraic Thinking | |
Concepts | • Patterns follow rules• Variables represent unknown quantities or numbers.• Estimation is an important strategy |
Skills | • Can generate a number/shape pattern that follows a stated rule. Can extend patterns• Can solve problems that include variables• Can use estimation to determine reasonableness of answers
• Can find the missing number in a multiplication or division equation |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • Understand that there are a variety of ways to measure: length, time, capacity, etc.• Understand that there are mathematical formulas to use when determining area and perimeter• Understand that there are mathematical formulas related to circles, triangles, angles
• Understands that capacity/volume can be measured |
Skills | • Can use protractors to determine degrees of angle, etc.• Can represent and interpret data• Use standard units of measurement to include inches, feet, centimeters, meters, etc.
• Can convert larger and smaller units • Can use measurement tools appropriately • Can use digital and analog clocks to measure elapsed • Can determine the area and perimeter of regular and irregular shapes • Can apply formulas to determine the area, perimeter, radius, diameter, circumference, and volume of shapes |
Data | |
Concepts | • Information can be collected and organized (data)• Understand that there are factors that influence the likelihood of certain outcomes (probability)• Understand the real-world applications involved in data and probability |
Skills | • Be able to create and interpret a visual representation of a data set using an appropriate method.• Conduct a probability simulation and interpret the outcome |
Geometry | |
Concepts | • Shapes can be classified by their attributes• Understands the difference between 2-D and 3-D images• Understands shapes as a combination of angles and lines
• Shapes within categories can vary • Understands symmetry • Understands categories of lines and their properties • Understands coordinate system |
Skills | • Can sort shapes by attribute• Can identify different shapes that have the same attributes• Can find lines of symmetry for 2-D figures
• Can measure angles using a protractor and/or formulas — recognize it as additive • Can graph points on the coordinate plane to represent real world and mathematical problems |
Grade 5
Numbers in Base 10 | |
Concepts | • Understands that the six digits of a six digit number are hundred thousands, ten-thousands, thousands, hundreds, tens, and ones• Understands that numbers can be written in expanded and standard form• Understands how decimals decimals to 1/1000 relate to place value |
Skills | • Can round to any place between 1,000,000 and 1/1000• Can read, write, and count to 1,000,000• Can decompose numbers between 1,000,000 and thousandths by place value in multiple ways |
Addition | |
Concepts | • You can add parts of whole numbers (fractions and decimals) as well as whole numbers |
Skills | • Ability to estimate to determine the reasonableness of an answer when adding whole numbers, decimals, and fractions• Apply decimal understanding to add money• Adds fractions with like and unlike denominators, decimals, and whole numbers |
Subtraction | |
Concepts | • you can subtract parts of whole numbers as well as whole numbers (fractions and decimals) |
Skills | • ability to estimate to determine reasonableness of answer when subtracting whole numbers, decimals, and fractions• apply decimal and place value understanding when subtracting money• can subtract fractions with like and unlike denominators, decimals and whole numbers |
Multiplication | |
Concepts | • Understands the patterns used when multiplying by a multiple of 10 (including 1/10, 1/100, and 1/1000)• Understands multiplication as a way to “scale something up or down”• Understands that multiplying by a number is the same as dividing by its reciprocal |
Skills | • Can fluently and accurately multiply any whole number• Can multiply by a multiple of 10 including 1/10, 1/100, 1/1000 |
Division | |
Concepts | • Understand that a fraction is a division problem• Understand how dividing by a multiple of 10 affects a decimal• Understand that a remainder can be written as a fraction or a decimal |
Skills | • Can find whole number quotients of whole numbers with up to 4-digit dividends and 2-digit divisors using some strategy• Can divide numbers by multiples of 10 through thousandths• Can write a remainder as a fraction or a decimal
• Can divide fractions and decimals |
Fractions | |
Concepts | • Deepen understanding of equivalencies through the application of all operations• Understand concepts of factors, multiples, GCF, LCD, LCM as related to fractions• Interpret a fraction as division of the numerator by the denominator
• Understand the relationship of fractions to decimals |
Skills | • Can decompose fractions into parts with unlike denominators• Add and subtract fractions with unlike denominators• Can multiply and divide fractions to include mixed numbers and improper fractions
• Can reduce fractions to lowest terms • Can solve story problems involving addition and subtraction of fractions, including unlike denominators • Can convert fractions into decimals |
Decimals | |
Concepts | • Understands that the first three places to the right of the decimal point are tenths, hundredths, and thousandths• Understands that decimals and fractions are both ways of expressing parts of a whole and can be used interchangeably |
Skills | • Can add, subtract, multiply, and divide decimals to the hundredths place, using any strategy• Can read, write, and compare decimals to thousandths• Can convert between common fractions (fourths, fifths, thirds, sixths, eighths and tenths) and decimals |
Percents | |
Concepts | • Understands that percent means parts out of 100• Understands that percent, like fractions and decimals, is another way to show part of a whole |
Skills | • Can use percents to describe common amounts (25%, 50%, 75%, 100%) |
Algebraic Thinking | |
Concepts | • Understands the rules of order of operations• Understands that mathematical thinking can be expressed using language, symbols, or a combination of both.• Understands the patterns used to create ordered pairs |
Skills | • Can identify ordered pairs and plot them on a coordinate plane• Can solve equations using the rules of order of operations including parentheses, exponents (squared), and four operations• Can express mathematical equations in either words or symbols (ex. 3 more than 7×8 = 3+(7×8)) |
The Nature of Mathematics | |
Concepts | • Recognize that math can be found everywhere (math drives technology, math models explain natural phenomena)• Appreciate the evolution of mathematics |
Mathematical Practices | • flexible, accurate and efficient with problem solving• look for and make use of patterns and models• uses math understanding to determine the reasonableness of answers
• able to defend and critique answers and strategies • can select and use appropriate tools • persevere with problem solving with flexibility • can apply mathematics to solve problems in everyday life |
Measurement | |
Concepts | • Understand that there are a variety of ways to measure: length, time, capacity, etc.• Understand that there are mathematical formulas to use when determining area and perimeter• Understand that there are mathematical formulas related to circles, triangles, angles
• Understands that capacity/volume can be measured |
Skills | • Can use protractors to determine degrees of angle, etc.• Can represent and interpret data• Use standard units of measurement to include inches, feet, centimeters, meters, etc.
• Can convert larger and smaller units • Can use measurement tools appropriately • Can use digital and analog clocks to measure elapsed • Can determine the area and perimeter of regular and irregular shapes • Can apply formulas to determine the area, perimeter, radius, diameter, circumference, and volume of shapes |
Data | |
Concepts | • Information can be collected and organized (data)• Understand that there are factors that influence the likelihood of certain outcomes (probability)• Understand the real-world applications involved in data and probability |
Skills | • Be able to create and interpret a visual representation of a data set using an appropriate method.• Conduct a probability simulation and interpret the outcome |
Geometry | |
Concepts | • Shapes can be classified by their attributes• Understands the difference between 2-D and 3-D images• Understands shapes as a combination of angles and lines
• Shapes within categories can vary • Understands symmetry • Understands categories of lines and their properties • Understands coordinate system |
Skills | • Can sort shapes by attribute• Can identify different shapes that have the same attributes• Can find lines of symmetry for 2-D figures
• Can measure angles using a protractor and/or formulas — recognize it as additive • Can graph points on the coordinate plane to represent real world and mathematical problems |