K-6 Elementary Mathematics Core Curriculum in Table Format st nd rd th th th Kindergarten 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade Standard 1: Standard 1: Standard 1: Standard 1: Standard 1: Standard 1: Standard 1: Students will Students will Students will Students will Students will Students will Students will understand acquire number acquire number understand the acquire number expand number expand number simple number sense and sense with base-ten sense and sense to include sense to include concepts and perform simple whole numbers numeration perform integers and operations with relationships. operations with and fractions system, place operations with perform rational whole numbers. and value whole operations with numbers. perform concepts, simple numbers, simple whole numbers, operations with fractions and fractions, and simple fractions, whole numbers. perform decimals. and decimals. operations with whole numbers. Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Identify and use Represent and Identify and Represent whole Demonstrate Represent whole Represent rational whole numbers use whole represent the numbers up to multiple ways to numbers and numbers in a variety up to 30. numbers up to relationships 10,000, represent whole decimals from of ways. 100. among numbers, comprehend numbers and thousandths to quantities, and place value decimals, from one billion, place value concepts, and hundredths to one fractions, in whole numbers identify million, and percents, and up to 1000. relationships fractions. integers. among whole numbers using base-ten models and symbolic notation. a. Represent a. Count, read, a. Represent a. Read, write, a. Read and write a. Read and write a. Recognize a whole numbers and write whole whole numbers in and represent numbers in numbers in rational number as a using concrete, numbers. groups of whole numbers standard and standard and ratio of two integers, pictorial, and hundreds, tens, using standard expanded form. expanded form. a to b, where b is not symbolic and ones using and expanded equal to zero. representations. base ten models form. and write the numeral representing the set in standard and expanded form. b. Order a set of b. Represent b. Identify the b. Demonstrate b. Demonstrate b. Demonstrate b. Change whole up to ten objects whole numbers place and the multiple ways to multiple ways to multiple ways to numbers with and use ordinal using the number value of a given represent represent whole represent whole exponents to numbers from first line, models, and digit in a three- numbers using numbers and numbers, standard form (e.g., to tenth to identify number digit numeral. models and decimals by using decimals, 24 = 16) and the sentences. symbolic models and fractions, recognize that position of the representations symbolic percents, and any non-zero whole object in the (e.g., fifty is the representations integers using number to the zero chosen order. same as two (e.g., 36 is the models and power equals 1 (e.g., groups of 25, the same as the symbolic 9^0 = 1). number of square of six, representations pennies in five three dozen, or 9 (e.g., 108 = 2 x 50 dimes, or 75 - 25). x 4). + 8; 108 = 102 + 8; 90% = 90 out of 100 squares on a hundred chart). c. Use one-to-one c. Represent c. Represent the c. Identify the c. Identify the c. Identify, read, c. Write a whole correspondence whole numbers composition and place and the place and the and locate number in expanded when counting a greater than 10 in decomposition of value of a given value of a given fractions, mixed form using set of objects and groups of tens numbers in a digit in a four-digit digit in a six-digit numbers, exponents (e.g., develop a and ones using variety of ways. numeral and numeral, including decimals, and 876,539 = 8 x 10^5 strategy for objects, round numbers decimals integers on the +7x keeping track of pictures, and to the nearest ten, to hundredths, number 10^4 + 6 x 10^3 + 5 counted and expanded hundred, and and round to the line. x 10^2 + 3 x 10^1 + uncounted notation. thousand. nearest tenth. 9 x 10^0). objects. d. Compare and d. Order and d. Divide regions, d. Represent d. Express numbers order numbers compare whole lengths, and sets repeated factors in scientific notation using the terms, numbers on a of objects into using exponents. using positive greater than, less number line and equal parts using powers of ten. than, or equal to, use the symbols a variety of and the <, >, ≠, and = models and symbols, >, <, when comparing illustrations. and =, using whole numbers. various strategies, including the number line. e. Identify and e. Identify factors e. Name and write e. Describe describe even and multiples of a fraction to situations where and odd whole whole numbers. represent a integers could be numbers. portion of a unit used in the whole, length, or students’ set for halves, environment. thirds, fourths, fifths, sixths, eighths, and tenths. f. Identify and represent square numbers using models and symbols. Objective 2: Objective 2: Objective 2: Use Objective 2: Use Objective 2: Objective 2: Objective 2: Explain Identify and use Identify simple unit fractions to fractions to Analyze Explain relationships and simple relationships identify parts of describe and relationships relationships and equivalencies relationships among whole the whole and compare parts of among whole equivalencies among rational among whole numbers up to parts of a set. the whole. numbers, among integers, numbers. numbers up to 30. 100. commonly used fractions, fractions, and decimals, and decimals to percents. hundredths. a. Estimate a. Compare and a. Divide a. Identify the a. Compare the a. Compare a. Place rational quantities in a set order sets of geometric shapes denominator of a relative size of fractions by numbers on the of objects using objects and into two, three, or fraction as the numbers (e.g., finding a common number line. multiples of 10 as numbers using four equal parts number of equal 475 is comparable denominator. benchmark the terms greater and identify the parts of the unit to 500; 475 is numbers. than, less than, parts as whole and small and halves, thirds, or the numerator of a compared to equal to when fourths. fraction as the 10,000 but large describing the number of equal compared to 98). comparisons. parts being considered. b. Compose and b. Make b. Divide sets of b. Define regions b. Order whole b. Order integers, b. Compare and decompose reasonable objects into two, and sets of numbers up to six fractions order rational quantities to estimates of the three, or four objects as a digits, simple (including mixed numbers, including establish a quantitative parts of equal whole and divide fractions, and numbers), and positive and relationship difference number of objects the whole into decimals using a decimals using a negative mixed between the parts between two sets and identify equal parts using variety of variety of fractions and and the of objects. the parts as a methods (e.g., methods, decimals, using a whole. halves, thirds, or variety of objects, number line, including the variety of methods fourths. models, and fraction pieces) number line. and symbols, illustrations. and use the including the number symbols <, >, and line and finding = to record the common relationships. denominators. c. Recognize 5 or c. Identify one c. Represent the c. Name and write c. Identify a c. Rewrite mixed c. Find equivalent 10 as a part of the more, one less, unit fractions 1/2, a fraction to number that is numbers and forms for common part-whole 10 more, and 10 1/3, and 1/4 with represent a between two improper fractions fractions, decimals, relationship of less than a given objects, pictures, portion of a unit given numbers from one form to percents, and ratios, numbers. number. words (e.g., whole for halves, (e.g., 3.2 is the other and including ___out of thirds, fourths, between 3 and 4; represent repeating or ___ equal parts), sixths, and find a each using terminating and symbols. eighths. number between regions, sets of decimals. 0.1 and 0.2). objects, or line segments. d. Compare sets d. Identify d. Place fractions d. Identify d. Represent d. Relate percents of objects and numbers missing on the number equivalences commonly used less than 1% or determine from a counting line and compare between fractions fractions as greater than 100% to whether they sequence. and order and decimals by decimals and equivalent fractions, have the same, fractions using connecting percents in a decimals, whole fewer, or more models, pictures, models to variety of ways numbers, and mixed objects. the number line, symbols. (e.g., numbers. and symbols. models, fraction strips, pictures, calculators, algorithms). e. Represent part- e. Find equivalent e. Generate e. Model and e. Recognize that whole fractions using equivalent calculate the sum of an relationships concrete and fractions and equivalent forms integer and its using the number pictorial simplify fractions of a fraction additive inverse is line. representations. using models, (including zero. pictures, and simplest form). symbols. f. Rename whole numbers as fractions with different denominators (e.g., 5 = 5/1, 3 = 6/2, 1 = 7/7). Objective 3: Objective 3: Objective 3: Objective 3: Objective 3: Objective 3: Use Objective 3: Use Model, describe, Model, describe, Estimate, model, Model problems Model and number theory number theory and illustrate and illustrate the illustrate, involving addition, illustrate concepts to concepts to find meanings of meanings of describe, and subtraction, meanings of develop and use prime factorizations, addition and addition and solve problems multiplication, and multiplication and divisibility tests; least common subtraction for subtraction and involving two- and division. division of whole classify whole multiples, whole use threedigit numbers and numbers to 50 as and greatest numbers less these operations addition and the addition and prime, composite, common factors. than ten. to solve subtraction. subtraction of or neither; and problems. fractions. find common multiples and factors. a. Demonstrate a. Use a variety of a. Demonstrate a. Demonstrate a. Model a. Identify a. Determine the joining and models, including quick recall of the meaning of multiplication patterns with skip whether whole separating of sets objects, length- addition facts (up multiplication and (e.g., equal-sized counting and numbers to 100 are of objects to solve based models, to 10 + 10) and division of whole groups, multiples to prime, composite, or problems. the number line related numbers through rectangular develop and use neither. and the subtraction facts. the use arrays, area divisibility tests for ten frame to of a variety of models, equal determining describe problem representations intervals on the whether a whole types (i.e., part- (e.g., equal-sized number line), number is whole, combine, groups, arrays, place value, and divisible by 2, 3, separate, area models, and properties of 5, 6, 9, and 10. compare). equal operations to jumps on a represent number line for multiplication of a multiplication, one- or two-digit partitioning and factor by a two- sharing for digit factor and division). connect the representation to an algorithm. b. Describe the b. Use the b. Model addition b. Use a variety of b. Use rectangular b. Use strategies b. Find the prime joining or properties of and subtraction of strategies and arrays to interpret for classifying factorization of separating of sets addition (i.e., two- and three- tools, such as factoring (e.g., whole numbers to composite numbers with informal commutativity, digit whole repeated addition find all rectangular 50 as prime, to 100. language when associativity, numbers (sums or subtraction, arrays of 36 tiles composite, or using models. identity element) and equal jumps and neither. and the minuends to on the number relate the mathematical 1000) in a variety line, and counters dimensions of the relationship of ways. arranged in arrays arrays to factors between addition to model of 36). and subtraction to multiplication and solve problems. division problems. c. Record c. Compute basic c. Write a story c. Demonstrate, c. Demonstrate c. Rewrite a c. Find the greatest pictorially the addition facts (up problem that using objects, that the mathematical composite common factor and results from to 10 + 10) and relates to a given multiplication and relationship number between least common joining or the related addition or division by the between 2 and 50 as a multiple for two separating of subtraction facts subtraction same number are multiplication and product of only numbers using a sets. using equation, and inverse operations division (e.g., 3 x prime numbers. variety of methods strategies (e.g., 6 write a (e.g., 3 x □ = 12 is □ = 12 is the (e.g., list of + 7 = (6 + 4) + 3 = number sentence the same as 12 ÷ same as 12 ÷ 3 = multiples, prime 10 + 3 = 13). to solve a story 3 = □ and □ = 4). □ and □ = 4) and factorization). problem that is use that related to the relationship to environment. explain that division by zero is not possible. d. Find the sum of d. Demonstrate d. Demonstrate d. Represent d. Find common three one-digit fluency with two- the effect of place division of a three- multiples and numbers. and three-digit value when digit dividend by a factors and apply addition and multiplying whole one-digit divisor, to adding and subtraction numbers by 10. including whole subtracting problems, using number fractions. efficient, accurate, remainders, using and generalizable a variety of strategies that methods (e.g., include standard rectangular algorithms and arrays, mental arithmetic, manipulatives, and describe why pictures), and the procedures connect the work. representation to an algorithm. e. Use the e. Write a story e. Use models to mathematical problem that add and subtract relationship relates to a given simple fractions between addition addition, where one single- and subtraction subtraction, or digit denominator and properties of multiplication is 1, addition to model equation, and 2, or 3 times the and solve write a number other (e.g., 2/4 + problems. sentence to solve 1/4; 3/4 - 1/8). a problem related to the students’ environment. Objective 4: Objective 4: Objective 4: Solve Objective 4: Objective 4: Model Model, illustrate, Compute and problems Model and and illustrate and pictorially solve problems involving illustrate meanings of record solutions involving addition multiplication and meanings of operations and to simple and subtraction of division of whole multiplication and describe how they multiplication and 3- and 4-digit numbers and division. relate. division problems. numbers and addition and basic facts of subtraction of multiplication and simple fractions division. and decimals. a. Represent a. Use a variety of a. Use estimation, a. Represent a. Relate fractions to multiplication with methods to mental math, division-with- multiplication and equal groups facilitate paper and pencil, remainder using division and use this using concrete computation (e.g., and calculators to whole numbers, relationship to objects and skip estimation, mental perform decimals, or explain counting by math strategies, mathematical fractions. procedures for twos, fives, and paper and pencil). calculations and multiplying and tens. identify when to dividing fractions. use each one appropriately. b. Represent b. Find the sum or b. Select b. Describe the b. Recognize that division as fair difference of appropriate effect of place ratios derive from shares using numbers, methods to solve value when pairs of rows in the concrete objects including a single operation multiplying and multiplication table or pictures. monetary problem and dividing whole and connect amounts, using estimate numbers and with equivalent models and computational decimals by 10, fractions. strategies such as results or 100, and 1,000. expanded form, calculate them compensation, directly, partial sums, and depending on the the standard context and algorithm. numbers involved in a problem. c. Compute basic c. Write a story c. Model c. Give mixed multiplication facts problem that multiplication of number and decimal (0-10) and related relates to a given fractions and solutions to division division facts multiplication or decimals (e.g., problems with whole using a variety of division equation, tenths multiplied numbers. strategies based and select by tenths, a whole on properties of and write a number multiplied addition and number sentence by tenths, or a multiplication (i.e., to solve a problem whole number commutative, related to the with tenths associative, environment. multiplied by identity, zero, and tenths) in a the distributive variety of ways properties). (e.g., manipulatives, number line and area models, patterns). d. Solve problems involving simple fractions and interpret the meaning of the solution (e.g., A pie has been divided into six pieces and one piece is already gone. How much of the whole pie is there when Mary comes in? If Mary takes two pieces, how much of the whole pie has she taken? How much of the pie is left?) Objective 5: Objective 5: Solve Objective 5: Solve Compute problems problems involving problems involving one or multiple steps. involving two operations. multiplication and division of whole numbers and addition and subtraction of simple fractions and decimals. a. Demonstrate a. Determine a. Select appropriate quick recall of when it is methods to solve a basic appropriate to use multi-step problem multiplication and estimation, mental involving division facts. math strategies, multiplication and paper and division of fractions pencil, and and decimals. algorithms. b. Multiply up to a b. Make b. Use estimation to three- digit factor reasonable determine whether by a two-digit estimations of results obtained factor with fraction and using a calculator fluency, using decimal sums, are reasonable. efficient differences, and procedures. products, including knowing whether results obtained using a calculator are reasonable. c. Divide up to a c. Write number c. Use estimation or three-digit sentences that calculation to dividend by a one- can be used to compute results, digit divisor with solve a two-step depending on the fluency, using problem. context and numbers efficient involved in the procedures. problem. d. Add and d. Interpret d. Solve problems subtract decimals division-with- involving ratios and and simple remainder proportions. fractions where problems as they one single-digit apply to the denominator is 1, environment (e.g., 2, If there or 3 times the are 53 people, other (e.g., 2/4 + how many vans 1/4 = 3/4; 1/3 – are needed if 1/6 = 1/6). each van holds 8 people?). Objective 6: Objective 6: Demonstrate Demonstrate proficiency with proficiency with the multiplication and four operations, with division of whole positive rational numbers and numbers, compute and with addition problems and subtraction of involving addition, integers. subtraction, and multiplication of decimals and fractions. a. Multiply multi- a. Multiply and divide digit whole a multi-digit number numbers by a by a two-digit two-digit whole number, including number with decimals. fluency, using efficient procedures. b. Divide multi- b. Add, subtract, digit dividends by multiply, and divide a one-digit divisor fractions and mixed with fluency, numbers. using efficient procedures. c. Add and c. Add and subtract subtract decimals integers. with fluency, using efficient procedures. d. Add and subtract fractions with fluency. e. Multiply fractions. Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Language and Language and Language and Language and Language and Language and Language and Symbols Symbols Symbols Symbols Symbols Symbols Symbols Students Students Should Students Should Students Should Students Should Students Should Students Should Should Use Use Use Use Use Use Use prime, composite, add, subtract, add, sum, number line, add, sum, difference, sum, difference, prime, composite, exponent, least first, second, subtract, sum, subtract, expanded form, expanded form, exponent, common multiple, third, fourth, fifth, difference, difference, greater factor, product, standard form, fractions, least common sixth, seventh, greater than, less than, less than, array, multiple, square number, numerator, denominator, eighth, ninth, than, equal to equal to, >, <, =, numerator, dividend, divisor, denominator, greatest common tenth, same, even, odd, denominator, quotient, factor, common factor, decimals, fewer, more halves, thirds, halves, product, array, denominator, percents, divisible, fourths, 1/2, 1/3, thirds, fourths, multiple, common divisibility, equivalent 1/4. sixths, eighths, numerator, factor, common fractions, integer, divisor, dividend, denominator, multiple, dividend, quotient, quotient, greater sixths, eighths, decimals, divisor, factor, than, less than, tenths, equivalent, percents, simplest terms, equal to, <, >, = estimate, <, >, divisible, mixed numeral, =, ≠ divisibility, improper fraction equivalent fractions, integer, dividend, quotient, divisor, factor, order of operations, simplest terms, various symbols for multiplication and division, mixed numeral, improper fraction Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Skills Skills Skills Skills Skills Skills Skills ��Count by ones, ��Use concrete ��Investigate ��Extend ��Use concrete ��Extend ��Explore the beginning from materials to addition of multiplication and objects and visual classification of addition and any number in the investigate common fractions division to larger- models to add whole numbers subtraction of counting situations that (e.g., ½ + ½ = 1, digit numbers. and subtract from 0-100 as positive and sequence. lead to ¼ + ¼ = ½ ). ��Use concrete common prime, composite, negative fractions. ��Represent multiplication and ��Investigate objects and visual decimals. or neither. ��Investigate the quantities using division. comparing models to add ��Explore ��Apply rules of concepts of ratio and concrete objects ��Develop and fractions in terms and subtract numbers less than divisibility. proportion. and investigate use strategies for of greater than, common zero by extending ��Explore adding ��Investigate the partitioning of addition and less than, and decimals. the number line and subtracting distributive property sets. subtraction of equal to. ��Investigate the and by using integers. of multiplication over ��Create multi-digit whole ��Understand distributive familiar ��Divide multi- addition of double- problems that can numbers. situations that property of applications such digit dividends by digit be solved using ��Investigate the entail multiplication over as temperature. a two-digit divisor. multipliers addition and meaning of multiplication and addition for single- ��Investigate the subtraction. fraction concepts. division, such as digit concept of ratio ��Understand equal groupings multipliers (e.g., 7 (e.g., the number situations that of x 15 is equivalent of students to the entail objects and to 7 x (10 + 5) is number of multiplication and sharing equally. equivalent to (7 x teachers). division, such as 10) + (7 x 5). equal groupings of objects and sharing equally. Standard 2: Standard 2: Standard 2: Standard 2: Standard 2: Standard 2: Standard 2: Students will Students will Students will Students will use Students will use Students will Students will use sort and classify identify and use model, patterns, patterns and use patterns and patterns, relations, objects as well number patterns represent, and symbols, relations to relations to and algebraic as recognize and and properties interpret operations, and represent represent and expressions to create simple to describe and patterns and properties of mathematical analyze represent patterns. represent number addition problems mathematical and analyze mathematical relationships to and and number problems and mathematical relationships. create and solve multiplication to relationships. number problems and problems with represent and relationships number addition and describe simple using algebraic relationships. subtraction. number symbols. relationships. Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Analyze Identify, sort, and Recognize, Recognize, Create, represent, Identify, analyze, Identify, analyze algebraic classify objects describe, and describe, create, and analyze and determine and determine a expressions, tables, according to represent and extend growing patterns. rules for rule for predicting and graphs to common patterns with growing patterns. describing and extending determine patterns, attributes. more than one numerical numerical relations, attribute. patterns involving patterns involving and rules. operations and operations whole nonnumerical numbers, growing patterns. decimals, and fractions. a. Sort objects a. Sort and a. Determine the a. Create and a. Analyze a. Analyze and a. Describe simple into groups by classify objects next term in linear extend growing growing patterns make predictions relationships by attribute and using more than patterns (e.g., 2, patterns using using objects, about numeric creating and identify which one attribute. 4, 6…; the objects, numbers, pictures, patterns, including analyzing tables, attribute was number of hands and tables. numbers, and decimals and equations, and used. on one tables to fractions. expressions. person, two determine a rule people, three for the pattern. people). b. Describe b. Identify, create, b. Construct b. Describe how b. Recognize, b. Determine a b. Draw a graph and multiple ways to and label models and skip patterns are represent, and rule for the write an equation sort and classify a repeating patterns count by twos, extended using extend simple pattern using from a table of group of objects. using objects, threes, fives, and manipulatives, patterns involving organized lists, values. pictures, and tens and relate to pictures, and multiples and tables, objects, symbolic notation. repeated numerical other number and variables. addition. representations. patterns (e.g., square numbers) using objects, pictures, numbers, and tables. c. Identify, create, c. Identify simple c. Draw a graph and and label growing relationships in create a table of patterns using real-life contexts values from an objects, pictures, and use equation. and symbolic mathematical notation. operations to describe the pattern (e.g., the number of legs on a given number of chairs may be determined by counting by fours or by multiplying the number of chairs by 4). d. Use patterns to establish skip counting by twos, fives, and tens. Objective 2: Objective 2: Objective 2: Objective 2: Objective 2: Use Objective 2: Use Objective 2: Write, Identify, duplicate, Recognize and Model, represent, Recognize, algebraic algebraic interpret, and use describe, and represent and interpret represent, and expressions, expressions, mathematical extend simple mathematical number simplify simple symbols, and inequalities, or expressions, repeating and relationships relationships number properties of the equations to equations, and growing patterns. using symbols using relationships operations to represent and formulas to and use number mathematical using symbols, represent, solve simple represent and solve sentences with symbols. operations, and simplify, and solve real-world problems that operational properties. mathematical problems. correspond to given symbols to solve equations and situations. problems. inequalities. a. Identify and a. Recognize that a. Recognize that a. Represent a. Use the order a. Use properties a. Solve single describe simple ―=‖ indicates that ―≠‖ indicates a numerical of operations to and the order of variable linear repeating patterns the two sides of relationship in relationships as evaluate, simplify, operations equations using a with numbers and an equation are which the two expressions, and compare involving addition, variety of strategies. shapes. expressions of sides of the equations, and mathematical subtraction, the same inequality are inequalities. expressions multiplication, number. expressions of involving the four division, and the different numbers. operations, use of parentheses, and parentheses to the symbols <, >, compute with and = (e.g., 2x (4 - whole numbers, 1) + decimals, and 3; of the two fractions. quantities 7 - (3 - 2) or (7 - 3) - 2, which is greater?). b. Duplicate and b. Recognize that b. Recognize that b. Solve b. Express single- b. Use patterns, b. Recognize that extend simple ―+‖ indicates the symbols such as equations operation problem models, and expressions in repeating patterns joining of sets and x, �� or ��in an , involving situations as relationships as different forms can with numbers and that ―-‖ indicates addition or equivalent equations and contexts for be equivalent and shapes. the separation of subtraction expressions (e.g., solve the writing and rewrite an sets. equation 6 + 4 = Δ + 7). equation. solving simple expression represent equations and to represent a a number that will inequalities with quantity in a different make the whole number way. statement true. solutions (e.g., 6x = 54; x + 3 = 7). c. Describe c. Write and solve c. Use the c. Use the >, <, c. Recognize that c. Evaluate and simple growing number commutative and and = symbols to a symbol simplify expressions patterns with sentences from associative compare two represents the and formulas, shapes. problem properties of expressions same number substituting given situations addition to involving addition throughout an values for the involving addition simplify and equation or variables and calculations. subtraction (e.g., expression (e.g., 2x + 4; x = 2; subtraction, using 4 + 6 □ 3 + 2; 3 + (e.g., Δ + Δ = 8; therefore, 2 (2) + 4 = symbolic notation 5 □ 16 - 9). thus, Δ = 4). 8). for the missing value (e.g., △+ 4 = 7). d. Identify simple d. Create problem d. Recognize and d. Describe and patterns in the situations from use the use the environment. given number commutative, commutative, sentences associative, associative, involving addition distributive, and distributive, and and identity properties identity properties subtraction. of of addition and addition and multiplication, and multiplication, and the zero property the zero property of multiplication. of multiplication. Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Language and Language and Language and Language and Language and Language and Language and Symbols Symbols Symbols Symbols Symbols Symbols Symbols Students Students Should Students Should Students Should Students Should Students Should Students Should Should Use Use Use Use Use Use Use order of operations, sort, repeating sort, attribute, patterns, +, -, =, ≠ growing patterns, growing pattern, variety of symbols sequence, function, patterns, growing repeating expressions, order of for multiplication pattern, algebraic patterns patterns, growing equations, <, >, = operations, and division such expression, patterns, skip parentheses, approximately equal, count, number inequality, symbols for ≈, sentence, symbol, expression, multiplication notation for +, -, = equation, and ÷ exponents: 43 or associative fraction bar (/ or 4^3, a number in property, —) as division front of a variable commutative symbols; variable, indicates property, order of multiplication (e.g., distributive operations, 3y property, zero parentheses, means 3 times the property of inequality, quantity y), formula, multiplication, >, expression, generalization <, = equation, associative property, commutative property, distributive property Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Skills Skills Skills Skills Skills Skills Skills ��Explore skip ��Investigate ��Investigate ��Use concrete ��Use concrete ��Solve multi- ��Use physical counting by fives, situations with situations with materials to build materials to build step equations. models to tens, and twos. variables as variables as an understanding an understanding ��Construct and investigate and unknowns and as unknowns and as of equality and of equality and analyze tables describe how a quantities that quantities that inequality. inequality. involving change in one vary. vary. ��Explore ��Explore equivalent ratios. variable affects a properties of properties of second variable. equality in number equality in number ��Use models to sentences (e.g., sentences (e.g., develop when equals are when equals are understanding of added to equals, added to equals, slope as constant then the sums are then the sums are rate of change. equal; when equal; when ��Model situations equals are equals are with proportional multiplied by multiplied by relationships and equals, then the equals, then the solve problems. products are products are equal. equal). Standard 3: Standard 3: Standard 3: Standard 3: Standard 3: Standard 3: Standard 3: Students will Students will Students will Students will Students will Students will Students will use understand understand understand describe and understand use spatial spatial and logical basic geometry simple geometry simple geometry analyze attributes and reasoning to reasoning to and and and attributes of two- properties of recognize, recognize, measurement measurement measurement dimensional plane geometric describe, and describe, and concepts as well concepts as well concepts as well shapes. objects analyze analyze geometric as as collect, as collect, and spatial geometric shapes and collect and represent, and represent, and relationships. shapes and principles. organize data. draw draw principles. conclusions conclusions from data. from data. Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Objective 1: Identify Identify and Identify, describe, Describe, classify, Describe and Identify and Describe and analyze create simple and create simple and create compare describe attributes relationships attributes and geometric shapes geometric figures. geometric figures. attributes of two- of two- between two- and properties of and describe dimensional dimensional three-dimensional geometric shapes to simple spatial shapes. geometric shapes. shapes and solve relationships. analyze problems. attributes and properties of geometric shapes. a. Identify, name, a. Name, create, a. Describe and a. Identify, a. Name and a. Draw, label, a. Identify the describe, and and sort classify plane and describe, and describe lines that and describe line midpoint of a line draw circles, geometric plane solid geometric classify polygons are parallel, segments, rays, segment and the triangles, figures (i.e., figures (i.e., circle, (e.g., pentagons, perpendicular, lines, parallel center and rectangles, and circle, triangle, triangle, hexagons, and intersecting. lines, and circumference of a squares in various rectangle, square, rectangle, octagons). perpendicular circle. sizes and trapezoid, square, trapezoid, lines. orientations. rhombus, rhombus, parallelogram, parallelogram, hexagon). pentagon, hexagon, cube, sphere, cone) according to the number of sides and angles or faces, edges, and vertices. b. Combine b. Identify b. Compose and b. Identify b. Identify and b. Draw, label, b. Identify angles as shapes to create geometric plane decompose attributes for describe right, and define an vertical, adjacent, two-dimensional and solid figures shapes and classifying acute, obtuse, angle as two rays complementary, or objects (e.g., (i.e., circle, figures by triangles (e.g., two and straight sharing a supplementary and using a triangle triangle, substituting equal sides for the angles. common endpoint provide and square to rectangle, square, arrangements of isosceles triangle, (vertex). descriptions of these create a picture of trapezoid, smaller three equal sides terms. a house). hexagon, shapes for larger for the equilateral rhombus, shapes or triangle, right parallelogram, substituting larger angle for the right cube, sphere, shapes for triangle). cone) in the arrangements of students’ smaller shapes. environment. c. Use words to c. Compose and c. Compose and c. Identify c. Identify and c. Classify c. Develop and use describe position decompose plane decompose attributes for describe the triangles and the properties of and distance. and solid figures shapes and classifying radius and quadrilaterals and complementary and (e.g., make two figures and quadrilaterals diameter of a analyze the supplementary triangles from a describe the part- (e.g., parallel circle. relationships angles and the sum square) whole sides for the among the of the angles of a and describe the relationships, parallelogram, shapes in triangle to solve part-whole similarities, and right angles for each classification problems involving relationships, the differences. the rectangle, (e.g., a square is an unknown angle in attributes of the equal sides and a rectangle). a triangle or figures, and how right angles for quadrilateral. they are the square). different and similar. d. Investigate two- d. Identify right d. Identify and d. Relate and three- angles in describe figures pyramids and dimensional geometric figures, that have line right prisms to the shapes including or in appropriate symmetry and two-dimensional hexagons, objects, and rotational shapes (nets) trapezoids, determine symmetry. from which they spheres, whether were created. cubes, and cones. other angles are greater or less than a right angle. e. Identify properties and attributes of solids (i.e., right prisms, pyramids, cylinders, cones) and describe them by the number of edges, faces, and vertices as well as the types of faces. Objective 2: Objective 2: Objective 2: Objective 2: Objective 2: Objective 2: Objective 2: Identify and use Identify Identify and use Demonstrate the Specify locations Specify locations Visualize and measurable measurable units of measure, meaning of using grids and in a coordinate identify geometric attributes of attributes of iterate (repeat) congruence maps. plane. shapes after objects and units objects and units that unit, and through applying applying of measurement. of measurement, compare the transformations. transformations on a and use number of coordinate plane. appropriate iterations to the techniques and item being tools to determine measured. measurements. a. Identify clocks a. Identify the a. Identify and a. Demonstrate a. Locate a. Locate points a. Rotate a polygon and calendars as appropriate tools use measurement the effect of coordinates in the defined by about the origin by a tools that for measuring units to measure, reflection, first quadrant of a ordered pairs of multiple of 90° and measure time. length, weight, to the nearest translation, or coordinate grid. integers. identify the location capacity, unit, length (i.e., rotation using of the new temperature, and inch, objects. vertices. time. centimeter), weight in pounds, and capacity in cups. b. Identify a day, b. Measure the b. Estimate and b. Determine b. Give the b. Write an b. Translate a week, and month length of an measure length whether two coordinates in the ordered pair for a polygon either on a calendar and object using by iterating a polygons are first quadrant of a point in a horizontally or name the days of nonstandard units nonstandard or congruent by coordinate grid. coordinate plane vertically on a the week in order. and count the standard unit of reflecting, with integer coordinate grid and units using measure. translating, or coordinates. identify the groups rotating one location of the new of tens and ones. polygon to vertices. physically fit on top of the other. c. Identify c. Identify the c. Use different c. Locate regions c. Specify c. Reflect a polygon pennies, nickels, value of a penny, units to measure on a map of Utah. possible paths across either the x- dimes, and nickel, dime, the length of the between locations or y-axis and identify quarters as units quarter, and same object and on a coordinate the location of the of money. dollar, and recognize that the plane and new determine the smaller compare vertices. value of the unit, the more distances of a set of the same iterations needed the various paths. coins that total to cover a given 25¢ or less (e.g., length. a set of 5 nickels equals 25¢). d. Compare two d. Tell time to the d. Determine the d. Give the objects by hour and half- value of a set of regions of a measurable hour. up to five coins position on a map attributes (i.e., that total $1.00 or of Utah. length, weight) less (e.g., three and order several dimes, objects by one nickel, and measurable one penny equals attributes (i.e., 36¢). length, weight). e. Name the e. Tell time to the months of the quarter-hour and year and seasons sequence a series in order, and use of daily events by a calendar to time (e.g., determine the day breakfast of the week and at 7:00 a.m., date. school begins at 9:00 a.m, school ends at 3:00 p.m.). Objective 3: Objective 3: Objective 3: Objective 3: Collect and Collect, organize, Collect, record, Visualize and organize simple and represent organize, display, identify geometric data. simple data. and interpret shapes after numerical data. applying transformations. a. Pose questions a. Collect and a. Collect and a. Identify a and gather data represent data record data translation, about self and using tables, tally systematically, rotation, or a surroundings. marks, using a strategy reflection of a pictographs, and for keeping track geometric shape. bar graphs. of what has been counted. b. Organize data b. Describe and b. Organize and b. Recognize that obtained from interpret data. represent the 90°, 180°, 270°, sorting and same data in and 360° are classifying more than one associated, objects. way. respectively, with 1/4, 1/2, 3/4, and full turns. c. Organize, display, and label information, including keys, using pictographs, tallies, bar graphs, and organized tables. d. Describe data represented on charts and graphs and answer simple questions related to data representations. Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Mathematical Language and Language and Language and Language and Language and Language and Language and Symbols Symbols Symbols Symbols Symbols Symbols Symbols Students Students Should Students Should Students Should Students Should Students Should Students Should Should Use Use Use Use Use Use Use midpoint, circle, triangle, circle, triangle, inch, centimeter, polygon, attribute, parallel, perpendicular and circumference, rectangle, square, rectangle, square, pound, cup, circle, quadrilateral, perpendicular, parallel lines, complementary and Sunday, Monday, trapezoid, triangle, equilateral intersecting lines, rays, angles supplementary Tuesday, hexagon, rectangle, square, triangle, isosceles right angle, acute (acute, obtuse, angles, rotate, Wednesday, rhombus, trapezoid, triangle, right angle, obtuse right, straight), translate, reflect, Thursday, Friday, parallelogram, rhombus, triangle, angle, straight triangles transformation Saturday, penny, cube, sphere, parallelogram, pentagon, angle, (equilateral, nickel, dime, cone, pentagon, hexagon, circle, radius, isosceles, quarter, shorter, penny, nickel, hexagon, cube, octagon, parallel, diameter, line scalene, right, longer, above, dime, quarter, sphere, cone, right angle, symmetry, acute, obtuse), below, near, far, dollar, January, vertices, angle, reflect, translate, rotational vertex, vertices, between February, March, face, edge, rotate, slide, flip, symmetry, edge, face, April, May , June, weight, length, turn, congruent coordinate, first corresponding July, August, capacity quadrant, degree, angles, similar, September, translate, rotate, polygon, pyramid, October, reflect, right prism November, transformation December, winter, spring, summer, fall, data, value, graph, tally mark Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Exploratory Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Concepts and Skills Skills Skills Skills Skills Skills Skills ��Measure ��Compare ��Use verbal ��Explore line ��Analyze results ��Compare ��Use manipulatives objects using non- objects using instructions to symmetry and of transformations corresponding and technology to standard units. non-standard move within the rotational (e.g., translations, angles of two model geometric ��Identify the units. environment. symmetry. rotations, triangles and shapes. value of a penny, ��Interpret data ��Determine ��Investigate two- reflections) on determine ��Investigate nickel, dime, and from charts and simple dimensional twodimensional whether the tessellations. quarter. graphs. equivalencies of representations of shapes. triangles are ��Explore the angles ��Organize data measurements. three-dimensional ��Investigate two- similar. formed by in lists, tables, ��Conduct simple objects. dimensional ��Rotate a shape intersecting lines. and simple probability representations of around a fixed ��Identify and draw graphs. experiments. three-dimensional point and identify shapes and figures objects. the location of the from different new vertices. views/perspectives. ��Translate a polygon either horizontally or vertically on a coordinate grid and identify the location of the new vertices. ��Reflect a shape across either the x- or y-axis and identify the location of the new vertices. Standard 4: Standard 4: Standard 4: Standard 4: Students will Students will Students will Students will select and use describe determine area understand and appropriate units relationships of polygons and apply and among units of surface area and measurement tools measurement measure, use volume of and techniques tools to solve appropriate three- and problems. measurement dimensional find the tools, and use shapes. circumference and formulas to find area of a circle. area measurements. Objective 1: Objective 1: Objective 1: Objective 1: Select and use Describe Determine the Describe and find appropriate tools relationships area of polygons the circumference and units to among units of and apply to real- and area of a circle. estimate and measure for world problems. measure length, length, capacity, weight, and weight, capacity, time, and determine and perimeter of measurements of two-dimensional angles using figures. appropriate tools. a. Describe the a. Describe the a. Determine the a. Explore the part-whole relative size area of a relationship between relationships (e.g., among metric trapezoid by the the radius and 3 feet in a yard, a units of length composition and diameter of a circle foot is 1/3 of a (i.e., millimeter, decomposition of to the circle’s yard) between centimeter, rectangles, circumference to metric units of meter), triangles, and develop the formula length (i.e., between metric parallelograms. for circumference. centimeter, units of capacity meter), and (i.e., milliliter, among customary liter), and units of length between metric (i.e., units of weight inch, foot, yard), (i.e., gram, capacity (i.e., cup, kilogram). quart), and weight (i.e., pound, ounce). b. Measure the b. Describe the b. Determine the b. Find the length of objects relative size area of irregular circumference of a to the nearest among customary and regular circle using a centimeter, meter, units of capacity polygons by the formula. half- and quarter- (i.e., cup, pint, composition and inch, foot, quart, gallon). decomposition of and yard. rectangles, triangles, and parallelograms. c. Measure c. Estimate and c. Compare areas c. Describe pi as the capacity using measure capacity of polygons using ratio of the cups and quarts, using milliliters, different units of circumference to the and measure liters, cups, pints, measure within diameter of a circle. weight using quarts, and the same pounds and gallons, and measurement ounces. measure weight system (e.g., using grams and square feet, kilograms. square yards). d. Identify the d. Recognize that d. Decompose a number of angles are circle into a number minutes in an measured in of wedges and hour, the number degrees and rearrange the of hours in a day, develop wedges into a shape the number of benchmark angles that days in a year, (e.g., 45°, approximates a and the number of 60°, 120°) using parallelogram to weeks in a year. 90° angles to develop the formula estimate angle for the area of a measurement. circle. e. Describe e. Measure e. Find the area of a perimeter as a angles using a circle using a measurable protractor or angle formula. attribute of two- ruler. dimensional figures, and estimate and measure perimeter with metric and customary units. Objective 2: Solve Objective 2: Objective 2: Objective 2: Identify problems Recognize and Recognize, and describe involving describe area as describe, and measurable measurements. a measurable determine surface attributes of objects attribute of two- area and volume and units of dimensional of three- measurement, shapes dimensional and solve problems and calculate area shapes. involving measurements. measurement. a. Determine a. Quantify area a. Quantify a. Recognize that simple by finding the total volume by finding measurements are equivalences of number of same- the total number approximations and measurements sized units of area of same-sized describe how the (e.g., 30 inches = needed to fill the units of volume size of the unit 2 feet and 6 region without needed to fill used in measuring inches; 6 gaps or overlaps. the space without affects the precision. cups = 1½ quarts; gaps or overlaps. 90 min. = 1 hr. 30 min.). b. Compare given b. Recognize that b. Recognize that b. Convert units of objects according a square that is 1 a cube having a 1 measurement within to measurable unit on a side is unit edge is the the metric system attributes (i.e., the standard unit standard unit for and convert units of length, weight, for measuring measuring measurement capacity). area. volume within the customary expressed as a system. cubic unit. c. Solve problems c. Develop the c. Derive and use c. Compare a meter involving area formula for a the formula to to a yard, a liter to a perimeter. rectangle and determine the quart, and a connect it with the volume of a right kilometer to a mile. area model for prism with a multiplication. triangular or rectangular base. d. Determine d. Develop and d. Relate the d. Determine when it elapsed time in use the area formulas for the is appropriate to hours (e.g., 7:00 formula for a right areas of triangles, estimate or use a.m. to 2:00 p.m.). triangle by rectangles, or precise comparing with parallelograms to measurement when the formula for a the surface solving rectangle (e.g., area of a right problems. two of the same prism. right triangles makes a rectangle). e. Develop, use, e. Derive and use e. Derive and use and justify the the formula to the formula to relationships determine the determine the among area surface area of a surface area and formulas of right prism and volume of a cylinder. triangles and express parallelograms by surface area in decomposing and square units. comparing with areas of right triangles and rectangles. f. Determine possible perimeters, in whole units, for a rectangle with a fixed area, and determine possible areas when given a rectangle with a fixed perimeter. Mathematical Mathematical Mathematical Mathematical Language and Language and Language and Language and Symbols Symbols Symbols Symbols Students Students Should Students Should Students Should Should Use Use Use Use cylinder, radius, measure, unit, millimeter, area, volume, diameter, metric system, centimeter, meter, surface area, circumference, area, customary milliliter, liter, volume, right surface area, system, length, gram, kilogram, prism volume, π pound, ounce, cup, pint, quart, centimeter, meter, gallon, area, inch, perimeter foot, yard, capacity, weight, perimeter Exploratory Exploratory Exploratory Exploratory Concepts and Concepts and Concepts and Concepts and Skills Skills Skills Skills ��Determine the ��Investigate ��Investigate pi ��Investigate value of a perimeter of as the ratio of the volumes and surface combination of rectangles and circumference to areas of a variety of coins and bills. squares. the diameter of a three-dimensional ��Count back ��Investigate area circle. objects. change from a of trapezoids. ��Determine the single purchase. volume of a right prism with various bases. Standard 5: Standard 5: Standard 5: Standard 5: Students will Students will Students will Students will collect and interpret and construct, analyze, draw organize data to organize analyze, and conclusions, and make collected data to construct make predictions predictions and make reasonable based upon identify basic predictions, conclusions data and apply concepts of answer from basic concepts of probability. questions, and data and apply probability. describe basic basic concepts concepts of of probability. probability. Objective 1: Objective 1: Objective 1: Objective 1: Design Collect, organize, Collect, organize, Formulate and investigations to and display data and display data answer questions reach conclusions to make to answer using statistical using statistical predictions. questions. methods to methods to make compare data, inferences based on and data. propose and justify inferences based on data. a. Collect, read, a. Identify a a. Construct, a. Design represent, and question that can analyze, and investigations to interpret data be answered by display data using answer questions. using tables, collecting data. an appropriate graphs, and format (e.g., line charts, including plots, bar keys graphs, line (e.g., pictographs, graphs). bar graphs, frequency tables, line plots). b. Make b. Collect, read, b. Recognize the b. Extend data predictions based and interpret data differences in display and on a data display. from tables, representing comparisons to graphs, charts, categorical and include scatter plots surveys, and numerical data. and circle graphs. observations. c. Represent data c. Identify c. Compare two using frequency minimum and similar sets of data tables, bar maximum values on the same graph graphs, line plots, for a set of data. and compare two and stem and leaf graphs representing plots. the same set of data. d. Identify and d. Identify and d. Recognize that distinguish calculate the changing the scale between clusters mean, median, influences the and outliers of a mode, and range. appearance of a data set. display of data. e. Propose and justify inferences and predictions based on data. Objective 2: Objective 2: Objective 2: Apply Objective 2: Apply Identify basic Describe and basic concepts of basic concepts of concepts of predict simple probability. probability and justify probability. random outcomes. outcomes. a. Describe the a. Describe the a. Describe the a. Write the results results of events results of results of of a probability using the terms experiments experiments experiment as a ―certain,‖ ―likely,‖ involving random involving random fraction between ―unlikely,‖ and outcomes as outcomes using a zero and one, or an ―impossible.‖ simple ratios (e.g., variety of equivalent percent. 4 notations (e.g., 4 out of 9, 4/9). out of 9, 4/9). b. Conduct simple b. Conduct simple a. Describe the b. Compare probability probability results of experimental results experiments, experiments, with experiments with theoretical record possible and without involving random results (e.g., outcomes replacement, outcomes using a experimental: 7 out systematically, record possible variety of of 10 tails; and outcomes notations (e.g., 4 whereas, theoretical display results in systematically, out of 9, 4/9). 5 out of 10 tails). an organized way and display (e.g., chart, results in an graph). organized way. c. Use results of c. Use the results c. Express the c. Compare simple probability of simple likelihood of an individual, small experiments to probability outcome in a group, and large describe the experiments, with simple experiment group results of a likelihood of a and without as a value probability specific replacement, to between 0 and 1 experiment in outcome in the describe the (inclusively). order to more future. likelihood of a accurately estimate specific outcome the actual in the future. probabilities. Mathematical Mathematical Mathematical Mathematical Language and Language and Language and Language and Symbols Symbols Symbols Symbols Students Students Should Students Should Students Should Should Use Use Use Use data display, scatter data, table, chart, data, line plot, line data, minimum plot, circle graph, graph, frequency graph, bar graph, values, maximum scale, predict, justify, table, line plot, stem and leaf plot, values, mean, probability, pictograph, bar cluster, outlier, median, mode, experimental results, graph, likely, frequency table, average, range theoretical results certain, outcome, probability impossible outcome Exploratory Exploratory Exploratory Exploratory Concepts and Concepts and Concepts and Concepts and Skills Skills Skills Skills ��Predict ��Explore ��Explore the ��Investigate the outcomes of minimum and differences in notion of fairness in simple maximum values representing games. experiments. for a set of data. categorical and ��Explore mean, numerical data. median, mode, and range.