Grades K–5 Scope and Sequence Your #1 Choice for World-Class Mathematics! Deep Math Understanding

Grades K–5 Scope and Sequence Your #1 Choice for World-Class Mathematics! Deep Math Understanding. Real-World Problem Solvers. Grade K Grade 1 Grade 2

Grade 3 Grade 4 Grade 5

Math in Focus®: Singapore Math by Marshall Cavendish is the U.S. Edition of Singapore’s most widely used program.

2 Scope and Sequence Grades K–5

Key Differences and Distinguishing Characteristics

Articulated Sequence Preparation for Algebra Math in Focus answers the call for a coherent sequence of topics Math in Focus answers the call to prepare students for algebra. giving students time to master foundational topics, As recommended by the National Math Panel, the Math in Focus so that little repetition is required the next year. Thus, each grade sequence of topics emphasizes: level covers fewer topics but in more depth, and you won’t find all • Number sense, basic facts, and computation An early topics in every grade level. understanding of composition and decomposition of numbers is • “Missing topics” When a topic appears to be “missing,” you can developed in tandem with mastery of basic be assured that it is found in either an earlier or later grade level. facts and computation algorithms in Grades K–2. For example you will find calendar concepts in Grades K and 1, but • Fractions and proportional reasoning Significant time is not repeated in Grade 2. allocated for in-depth work with fractions in Grades 3–5. • More advanced As a result of not repeating topics year after year, • Problem solving Challenging problem solving is built students who use Math in Focus will advance faster than students in into each chapter in every grade level. other programs. As a result, you may find topics that seem to be “too advanced.” However, you will find your students easily able to handle the challenge as long as they have had the appropriate preliminary instruction.

Developmental Continuum Kindergarten Grades 1–2 Grades 3–5

Foundational concepts through songs, rhymes, and hands-on activities

Concept and skill development through hands-on instruction and practice • basic facts • place value • mental math • geometry concepts Emphasis on problem solving, skill consolidation, and a deep understanding in preparation for algebra

• fractions • ratios • expressions, • decimals • model drawing equations, and inequalities

3 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations Number and Operations Sets and Use concrete models to create Use concrete and pictorial Use concrete and pictorial Sets and Numbers a set with a given number of models to create a set with a models to create a set with Numbers objects given number of objects (up to a given number of objects (up to 20). 100). (up to 1,000). Group objects and numbers up Group objects and numbers up to 100 in tens and ones. to 1,000 into hundreds, tens, and ones. Group objects into equal sized groups.

Use cardinal and ordinal Use cardinal numbers up to Explore negative numbers numbers. 100 and ordinal numbers up in context. to 10th. Number Use numbers to represent Use number bonds to represent Use place value models to create Number Represent numbers to 10,000 Represent numbers to 100,000 Understand place value Representation quantities up to 20. number combinations. equivalent representations of Representation in different equivalent forms. in various contexts. concepts through millions. numbers.

Represent numbers to 100 on a Represent numbers to 1,000 on number line. a number line. Count Count up to 20 objects in a set. Count to 100. Count to 1,000. Count Count to 10,000.

Count on and back to 20. Count by 1s, 2s, 5s, and 10s Count by multiples of ones, Count by hundreds and Count by thousands and ten Count by hundred thousands forward and backward to 100. tens, and hundreds. thousands. thousands. and millions.

Count in 2s and 5s up to 20.

Compare and Compare and order sets and Compare and order whole Compare and order whole Compare and Compare and order whole Compare and order whole Compare and order whole Order numbers up to 20. numbers to 100. numbers to 1,000. Order numbers to 10,000. numbers to 100,000. numbers to 10,000,000.

Compare and order using the Compare and order using Use <, >, and = to compare terms fewer, more, and less. the terms same, more, fewer, whole numbers. greater than, less than, equal to, greatest, and least. Place Value Use place value models and Use base-ten models and Place Value Use place value models to Express numbers to 100,000 in Express numbers to 10,000,000 place value charts to represent place value charts to represent read, write, and represent standard, expanded, and word in various forms. numbers to 100. numbers to 1,000. numbers to 10,000. forms.

4 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations Number and Operations Sets and Use concrete models to create Use concrete and pictorial Use concrete and pictorial Sets and Numbers a set with a given number of models to create a set with a models to create a set with Numbers objects given number of objects (up to a given number of objects (up to 20). 100). (up to 1,000). Group objects and numbers up Group objects and numbers up to 100 in tens and ones. to 1,000 into hundreds, tens, and ones. Group objects into equal sized groups.

Represent numbers to 100 on a Represent numbers to 1,000 on number line. a number line. Count Count up to 20 objects in a set. Count to 100. Count to 1,000. Count Count to 10,000.

Count in 2s and 5s up to 20.

5 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Place Value Express numbers to 100 in Express numbers to 1,000 Place Value (continued) standard and word forms. in terms of place value. (continued)

Compose and decompose multi-digit numbers (including expanded form).

Fraction Connect geometric concepts Fraction Understand the meanings Recognize, write, name, and Convert fractions to decimals. Concepts with unit fractions—halves, Concepts and uses of fractions including illustrate mixed numbers and thirds, and fourths. fraction of a set. improper fractions.

Understand the relationship Understand that the size of a Find a fraction of a set. Relate fractions and between a fraction and a whole. fractional part is relative to the division expressions. size of the whole.

Compare and order halves, Compare fractions using Generate equivalent fractions. thirds, and fourths using models and number lines. bar models.

Identify equivalent fractions Convert among mixed through the use of models, numbers and multiplication, division, and improper fractions. number lines.

Add and subtract like fractions.

Money Identify and relate coin values Identify and relate coin values Identify $1, $5, $10, and Money Add and subtract money. (penny, nickel, dime, quarter). (penny, nickel, dime, quarter). $20 bills. Count and make Count and make Count and make combinations coin combinations. coin combinations. of coins and bills. Compare money amounts. Solve real-world problems involving addition and subtraction of money. Decimal Use the dollar sign and Decimal Use the dollar sign and decimal Model decimals using tenths Model decimals Concepts decimal point. Concepts point in money amounts. and hundredths. using thousandths.

6 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Place Value Express numbers to 100 in Express numbers to 1,000 Place Value (continued) standard and word forms. in terms of place value. (continued)

Compose and decompose multi-digit numbers (including expanded form).

Compare and order halves, Compare fractions using Generate equivalent fractions. thirds, and fourths using models and number lines. bar models.

Identify equivalent fractions Convert among mixed through the use of models, numbers and multiplication, division, and improper fractions. number lines.

Add and subtract like fractions.

7 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Decimal Decimal Understand decimal notation Understand place value Concepts Concepts through hundredths as an concepts through thousandths. (continued) (continued) extension of the base-ten system. Read and write decimals that are greater than or less than 1. Compare and order decimals. Identify equivalent decimals.

Connect equivalent fractions Convert decimals to fractions. and decimals. Ratio, Ratio, Use ratios to solve problems. Proportion, and Proportion, and Percent Percent Find equivalent ratios.

Solve problems with percent.

Convert fractions to percents. Find a percent of a number. Whole Number Model joining and separating Model addition and subtraction Model addition and subtraction Whole Number Model regrouping in addition Computation: sets. situations. with place value. Computation: and subtraction with place Addition and Addition and value. Subtraction Subtraction Use +, -, and = to write number Use models, numbers, and Recall addition and subtraction sentences for addition and symbols for addition and facts. subtraction stories. subtraction facts to 20. Use the order, grouping, and Use different methods to zero properties to develop develop fluency in adding addition and subtraction fact and subtracting multi-digit strategies. numbers. Add and subtract up to 2-digit Add and subtract whole Add and subtract whole numbers with and without numbers to 1,000. numbers to 10,000. regrouping

8 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Decimal Decimal Understand decimal notation Understand place value Concepts Concepts through hundredths as an concepts through thousandths. (continued) (continued) extension of the base-ten system. Read and write decimals that are greater than or less than 1. Compare and order decimals. Identify equivalent decimals.

Connect equivalent fractions Convert decimals to fractions. and decimals. Ratio, Ratio, Use ratios to solve problems. Proportion, and Proportion, and Percent Percent Find equivalent ratios.

Solve problems with percent.

9 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Whole Number Represent addition and Formulate addition and Whole Number Computation: subtraction stories. subtraction stories. Computation: Addition and Addition and Subtraction Subtraction Real-World Real-World Problems Problems Solve addition and subtraction Solve multi-digit addition Solve addition and subtraction problems using basic facts. and subtraction problems problems with greater by using a bar model. numbers by using a bar model. Whole Number Count by 2s and 5s up to 20. Count by 2s, 5s, and 10s. Multiply and divide with 2, Whole Number Multiply and divide with 6, 7, Computation: 3, 4, 5, and 10. Computation: 8, and 9. Multiplication Multiplication and Division and Division Concepts Concepts Adding the same number to Represent multiplication as Represent multiplication in Apply understanding of models multiply. repeated addition. different ways. for multiplication and division.

Represent sharing equally and Represent division as repeated Represent division in different Recall multiplication facts and making equal groups. subtraction. ways. related division facts. Use the ×, ÷, and = symbols to represent multiplication and division situations. Whole Number Whole Number Multiply 1s, 10s, and 100s with Develop fluency in multiplying Multiply multi-digit numbers. Computation: Computation: and without regrouping. multi-digit numbers. Multiplication Multiplication and Division Al- and Division Al- gorithms gorithms Use addition and multiplication properties to multiply.

Divide 10s and 1s with and Divide by a 1-digit number, Find quotients involving multi- without regrouping, no with a remainder. digit dividends. remainder.

10 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Whole Number Represent addition and Formulate addition and Whole Number Computation: subtraction stories. subtraction stories. Computation: Addition and Addition and Subtraction Subtraction Real-World Real-World Problems Problems Solve addition and subtraction Solve multi-digit addition Solve addition and subtraction problems using basic facts. and subtraction problems problems with greater by using a bar model. numbers by using a bar model. Whole Number Count by 2s and 5s up to 20. Count by 2s, 5s, and 10s. Multiply and divide with 2, Whole Number Multiply and divide with 6, 7, Computation: 3, 4, 5, and 10. Computation: 8, and 9. Multiplication Multiplication and Division and Division Concepts Concepts Adding the same number to Represent multiplication as Represent multiplication in Apply understanding of models multiply. repeated addition. different ways. for multiplication and division.

Divide 10s and 1s with and Divide by a 1-digit number, Find quotients involving multi- without regrouping, no with a remainder. digit dividends. remainder.

11 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Whole Number Use bar models to represent Whole Number Use bar models to represent Computation: multiplication and division Computation: multiplication and division Multiplication situations. Multiplication situations. and Division and Division Real-World Real-World Problems Problems Solve multiplication and division Solve one- and two-step Solve multi-digit multiplication Solve multiplication and division fact problems. multiplication and division and division problems. problems. problems.

Select the most useful form of the quotient and interpret the remainder. Fraction Add and subtract like fractions Fraction Add and subtract like fractions. Add and subtract unlike Add and subtract unlike fractions Computation (halves, thirds, fourths). Computation fractions. and mixed numbers.

Multiply proper fractions, improper fractions, mixed numbers, and whole numbers. Divide fractions by whole numbers. Solve word problems with addition, subtraction, multiplication, and division of fractions. Decimal Add and subtract money. Solve addition and subtraction Decimal Add and subtract money Add and subtract decimals. Add and subtract decimals. Computation money problems. Computation amounts. Multiply and divide decimals by whole numbers. Solve problems with addition Solve problems with and subtraction of decimals. multiplication and division of decimals. Estimation and Use mental math strategies to Use mental math strategies to Estimation and Use mental math strategies Use mental math and Use estimation and mental math Mental Math add and subtract. add and subtract. Mental Math to add subtract, multiply, and estimation strategies to find to estimate sums, differences, divide. sums, differences, products, products, and quotients. and quotients.

12 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Whole Number Use bar models to represent Whole Number Use bar models to represent Computation: multiplication and division Computation: multiplication and division Multiplication situations. Multiplication situations. and Division and Division Real-World Real-World Problems Problems Solve multiplication and division Solve one- and two-step Solve multi-digit multiplication Solve multiplication and division fact problems. multiplication and division and division problems. problems. problems.

13 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Estimation and Estimation and Decide whether an estimate or Mental Math Mental Math exact answer is needed. (continued) (continued) Estimate quantity by using Round to the nearest ten to Use front-end estimation and Use estimation in determining Estimate sums and differences referents. estimate sums and differences. rounding to estimate sums and relative sizes with fractions differences. of amounts or distances. and decimals.

Round and estimate with Estimate products and quotients decimals. with decimals. Algebra Algebra

Patterns Describe and extend repeating Identify, describe, and extend Describe, extend, and create Patterns Create and analyze Identify, describe, and extend Identify, describe, and extend shape patterns. two- and three-dimensional two-dimensional shape multiplication and division numeric and non-numeric numeric patterns involving shape patterns. patterns. patterns. patterns. all operations. Count by 2s and 5s. Skip count by 2s, 5s, and 10s. Skip count by 2s, 3s, 4s, 5s, and Skip count by 6s, 7s, 8s, and 9s. 10s.

Describe a rule for sorting Identify a rule for sorting objects. objects.

Find missing terms in repeating Identify and extend growing Identify rules for number Analyze number and counting Use a rule to describe a Find rules to complete number patterns. and repeating patterns. patterns. patterns. sequence of numbers or patterns. objects. Find missing terms in growing Find missing terms in table and repeating patterns. patterns.

Properties Identify 0 as the identity Understand that addition Properties Understand that multiplication Represent division as the element for addition and and subtraction are inverse and division are related. inverse of multiplication. subtraction. operations. Use the Associative and Apply properties of addition. Create and analyze Commutative Properties of multiplication and division Addition. patterns. Use the Distributive Property as Model, define, and explain a multiplication strategy. properties of multiplication.

Number Identify odd and even Number Identify odd and even Find the greatest common Theory numbers. Theory numbers. factor and least common multiple. Identify prime and composite numbers.

14 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Number and Operations (continued) Number and Operations (continued) Estimation and Estimation and Decide whether an estimate or Mental Math Mental Math exact answer is needed. (continued) (continued) Estimate quantity by using Round to the nearest ten to Use front-end estimation and Use estimation in determining Estimate sums and differences referents. estimate sums and differences. rounding to estimate sums and relative sizes with fractions differences. of amounts or distances. and decimals.

Round and estimate with Estimate products and quotients decimals. with decimals. Algebra Algebra

Describe a rule for sorting Identify a rule for sorting objects. objects.

Number Identify odd and even Number Identify odd and even Find the greatest common Theory numbers. Theory numbers. factor and least common multiple. Identify prime and composite numbers.

15 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Algebra (continued) Alegbra (continued) Functional Understand the relationships Recognize how bar models Functional Understand the relationships Understand the relationships Understand the relationships Relationships between the numbers in fact show relationships between Relationships between the numbers in between the numbers and between the numbers and families. numbers and unknowns in multiplication and division fact symbols in formulas for area symbols in formulas for surface number sentences. families. and perimeter. area and volume. Describe number relationships Describe number relationships Describe number relationships in context. in context. in context.

Expressions/ Use a variety of concrete, Use a variety of concrete, Expressions/ Use a variety of concrete, Use a variety of concrete, Use letters as variables. Models pictorial, and symbolic models pictorial, and symbolic models Models pictorial, and symbolic models pictorial, and symbolic models for addition and subtraction. for addition, subtraction, for multi-digit addition, for multiplication and division; multiplication, and division. subtraction, multiplication, and addition and subtraction and division. with fractions and decimals. Simplify algebraic expressions.

Use the order of operations in numeric expressions with two or more operations. Number Model addition and subtraction Model addition and subtraction Model multiplication and Number Write multiplication and Write and solve number Write and solve number Sentences stories with addition and situations by writing addition division situations by writing Sentences division number sentences. sentences for one-, two-, sentences and equations for and Equations subtraction number sentences. and subtraction number multiplication and division and Equations and three-step real-world one- and two-step real-world sentences. number sentences. problems. problems.

Use bar models and number Write and solve number Use bar models and number Write and solve equations. sentences to represent real- sentences for one- and two- sentences for one-, two-, world problems. step real-world problems. and three-step real-world problems. Determine the value of missing Determine the missing parts Determine the missing parts Graph linear equations. quantities in number sentences. (quantities or symbols) in (quantities or symbols) in number sentences. number sentences. Equality and In- Understand the meaning of the Understand the difference Use and create models that Equality and In- Understand equality and Understand equality and Understand equality and equality = sign in number sentences. between equality and demonstrate equality or equality inequality. inequality. inequality. inequality. inequality. Use <, >, and = to write number Write and solve inequalities. sentences.

Geometry Geometry Size and Understand big, middle- sized, Describe position with left and Size and Position and small. right. Position

16 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Algebra (continued) Alegbra (continued) Functional Understand the relationships Recognize how bar models Functional Understand the relationships Understand the relationships Understand the relationships Relationships between the numbers in fact show relationships between Relationships between the numbers in between the numbers and between the numbers and families. numbers and unknowns in multiplication and division fact symbols in formulas for area symbols in formulas for surface number sentences. families. and perimeter. area and volume. Describe number relationships Describe number relationships Describe number relationships in context. in context. in context.

Geometry Geometry Size and Understand big, middle- sized, Describe position with left and Size and Position and small. right. Position

17 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Geometry (continued) Geometry (continued) Size and Describe and compare objects Use positional words to Size and Position by position. describe location. Position (continued) (continued) Lines and Identify parts of lines and Lines and Identify perpendicular and Draw perpendicular and Work with angles on a straight Angles curves. Angles parallel lines. parallel lines. line. Identify right angles and Construct and measure angles. Work with angles at a point. compare angles to right angles. Two- Identify similarities and Two- Dimensional differences. Dimensional Shapes Shapes

Name flat shapes that make up Identify real-world two- real-world objects. dimensional shapes.

Identify, describe, sort, and Identify and describe attributes Identify, describe, sort, and Describe, analyze, compare, Apply the properties of squares Apply the properties of right, classify two-dimensional and properties of two- classify two-dimensional and classify two-dimensional and rectangles. isosceles, and equilateral shapes. dimensional shapes. shapes. shapes by their sides and triangles. angles. Sort and classify two- Identify parts of lines and Classify and sort polygons and Find unknown angle measures Apply the sum of the angle dimensional shapes. curves. quadrilaterals by attributes and and side lengths of squares and measures of a triangle. properties. rectangles. Make flat shape pictures. Compose and decompose two- Compose and decompose two- Investigate composing and Identify figures that form Apply the properties of a dimensional shapes. dimensional shapes. decomposing two-dimensional tessellations. parallelogram, rhombus, and shapes. trapezoid.

Use attributes and properties Demonstrate that the sum of any to solve problems. two side lengths of a triangle is greater than the length of the third side. Compare areas using Develop foundations for Find and compare the area Understand the relationships Find the area of a triangle. non-standard units. understanding area. of plane figures in different between the numbers and square units. symbols in formulas for area and perimeter. Three- Name and sort solid shapes. Identify real-world three- Three- Identify and classify prisms and Dimensional dimensional shapes. Dimensional pyramids. Shapes Shapes Understand that three- Identify two-dimensional Identify, describe, sort, and Identify the solid that can be dimensional shapes are made shapes in three-dimensional classify three-dimensional made from a net. up of two-dimensional shapes. shapes. shapes.

18 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Geometry (continued) Geometry (continued) Size and Describe and compare objects Use positional words to Size and Position by position. describe location. Position (continued) (continued) Lines and Identify parts of lines and Lines and Identify perpendicular and Draw perpendicular and Work with angles on a straight Angles curves. Angles parallel lines. parallel lines. line. Identify right angles and Construct and measure angles. Work with angles at a point. compare angles to right angles. Two- Identify similarities and Two- Dimensional differences. Dimensional Shapes Shapes

Name flat shapes that make up Identify real-world two- real-world objects. dimensional shapes.

19 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Geometry (continued) Geometry (continued) Three- Sort and classify three- Identify surfaces that slide, Three- Identify cylinders, spheres, and Dimensional dimensional shapes. stack, and roll. Dimensional cones. Shapes Shapes (continued) (continued) Recognize shapes from Describe cylinders, spheres, different perspectives. and cones by the number of and types of faces, and the number of edges and vertices. Compose and decompose Build solids using unit cubes. three-dimensional shapes.

Congruence Develop initial understanding Congruence Identify symmetrical figures Identify line and rotational and Symmetry of congruence and symmetry. and Symmetry and one line of symmetry. symmetry. Solve problems involving Relate rotational symmetry to congruency. turns and congruency.

Transformations Transformations Identify pairs of shapes that Use transformations to form show a flip, slide, and turn. tessellations.

Demonstrate that figures and their flip, slide, and turn images are congruent. Coordinate Coordinate Develop coordinate readiness Plot points on a coordinate grid. Geometry Geometry with tables and line graphs. Measurement Measurement Length and Dis- Compare lengths and heights Compare two lengths by Demonstrate linear measure as Length and Dis- Select appropriate units and tance using non-standard units. comparing each with a third an iteration of units. tance tools to estimate and measure length (transitivity). length.

Compare and order lengths Use a start line to measure Use rulers to measure length. Use meter sticks, 12-inch (long, short, longest, shortest). length. rulers, and yardsticks to measure length. Develop a background for Measure lengths, using Measure lengths in meters, Measure length to the nearest measurement using non-standard units. centimeters, feet, and inches. half inch and inch. non-standard units.

Explain the need for equal- Use referents to estimate length units to measure. distance.

20 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Geometry (continued) Geometry (continued) Three- Sort and classify three- Identify surfaces that slide, Three- Identify cylinders, spheres, and Dimensional dimensional shapes. stack, and roll. Dimensional cones. Shapes Shapes (continued) (continued) Recognize shapes from Describe cylinders, spheres, different perspectives. and cones by the number of and types of faces, and the number of edges and vertices. Compose and decompose Build solids using unit cubes. three-dimensional shapes.

Transformations Transformations Identify pairs of shapes that Use transformations to form show a flip, slide, and turn. tessellations.

Explain the need for equal- Use referents to estimate length units to measure. distance.

21 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Length and Dis- Explain the need for equal- Length and Dis- tance length units to measure. tance (continued) (continued) Count length units in groups of Compare and measure lengths Estimate and measure length, 10s and 1s. using customary distance, and height in meters, and metric units. centimeters, and kilometers.

Compare measurements made Demonstrate partitioning and Convert among metric units of using different units. transitivity in relation length. to length.

Understand the inverse Solve problems involving Solve one- and two-step relationship between the size of estimating, measuring, and real-world problems in a unit and the number of units. computing length. measurement. Weight/Mass Order objects by weight. Compare and measure weights Compare and measure masses. Weight/Mass Select appropriate units and using non-standard units. tools to estimate and measure weight. Compare weights using non- Compare two masses by Use referents to estimate standard units. comparing each with a third weight. mass (transitivity). Estimate and find masses of objects. Solve weight problems. Solve mass problems. Convert among units of mass.

Capacity/ Compare capacities using non- Measure volume (capacity) Capacity/ Select appropriate tools and Volume standard units. in liters. Volume units to estimate and measure volume and capacity. Solve volume problems. Determine the volume and capacity of a container.

Relate the units of customary capacity to one another.

Use referents to estimate capacity.

22 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Length and Dis- Explain the need for equal- Length and Dis- tance length units to measure. tance (continued) (continued) Count length units in groups of Compare and measure lengths Estimate and measure length, 10s and 1s. using customary distance, and height in meters, and metric units. centimeters, and kilometers.

Compare measurements made Demonstrate partitioning and Convert among metric units of using different units. transitivity in relation length. to length.

Relate the units of customary capacity to one another.

Use referents to estimate capacity.

23 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Capacity/ Capacity/ Estimate and measure capacity Volume Volume in liters and milliliters. (continued) (continued) Convert among metric units of capacity.

Time Name and order the days of the Read a calendar to identify the Time week and the months of the days of the week, months, and year. seasons of the year. Recognize the correct way to Use A.M. and P.M. to write time. Read time on a digital clock. write the date.

Tell time to the hour and half Tell time to five minutes. Convert between hours and hour. minutes.

Compare durations of events. Find elapsed time. Determine elapsed time.

Add and subtract units of time.

Temperature Temperature Read a Fahrenheit thermometer.

Choose the appropriate tool and unit to measure temperature. Use referents to estimate temperature.

Angles Angles Compare angles to right Estimate and measure angles Apply the idea that the sum of angles. with a protractor. angles on a straight line is 180°. Classify angles by angle Apply the idea that vertical measure. angles are equal in measure.

Relate 1/4-, 1/2-, 3/4-, and full Apply the idea that the sum of turns to the number of right angles at a point is 360°. angles. Perimeter Perimeter Measure perimeter of plane Find the perimeter of figures. composite figures.

24 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Capacity/ Capacity/ Estimate and measure capacity Volume Volume in liters and milliliters. (continued) (continued) Convert among metric units of capacity.

Tell time to the hour and half Tell time to five minutes. Convert between hours and hour. minutes.

Compare durations of events. Find elapsed time. Determine elapsed time.

Add and subtract units of time.

Temperature Temperature Read a Fahrenheit thermometer.

Choose the appropriate tool and unit to measure temperature. Use referents to estimate temperature.

25 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Perimeter Perimeter Choose the appropriate tool, Solve problems involving (continued) (continued) unit, and strategy to measure the perimeter of squares, perimeter. rectangles, and composite figures. Estimate the perimeter of surfaces and objects.

Area Compare areas using Compose and decompose two- Develop foundations for Area Find and compare the area Explain area as an attribute of non-standard units. dimensional shapes understanding area. of plane figures in different two-dimensional figures. (foundation for understanding square units. area).

Make different plane figures Connect area measure to the Find the area of triangles. with the same area. area model for multiplication; use it to justify the formula for the area of a rectangle. Estimate area of small and Estimate and measure area in large surfaces. square units.

Compare the area and Select appropriate units, perimeter of two plane figures. strategies, and tools to solve area problems. Find the area of rectangles and Explain the relationships composite figures. among area formulas of different polygons.

Surface Area Surface Area Decompose solid figures to and Volume and Volume find the surface area. Estimate and measure volume in cubic units.

Data Analysis Data Analysis Classifying and Understanding similarities Sort and classify geometric Sort and classify two- and three- Classifying and Classify and sort polygons and Sorting and differences in objects and shapes. dimensional shapes Sorting quadrilaterals by attributes and shapes. by properties. properties.

Sorting and classifying objects Sorting and classifying data in Collect and organize data in Collect and organize data in Construct line plots, stem- Represent data in a double bar using one or two attributes. order to make graphs. picture graphs. bar graphs and line plots. and-leaf plots, tables, and line graph. graphs.

26 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Measurement (continued) Measurement (continued) Perimeter Perimeter Choose the appropriate tool, Solve problems involving (continued) (continued) unit, and strategy to measure the perimeter of squares, perimeter. rectangles, and composite figures. Estimate the perimeter of surfaces and objects.

Surface Area Surface Area Decompose solid figures to and Volume and Volume find the surface area. Estimate and measure volume in cubic units.

27 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Data Analysis (continued) Data Analysis (continued) Collect and Organize data for a picture Collect and organize data in Collect and organize data in Collect and Organize Data graph. different ways. different ways. Organize Data

Represent Data Represent data in pictographs. Represent measurements and Represent data in picture Represent Data data in picture graphs, tally graphs. charts, and bar graphs.

Interpret/ Interpret data in tally charts and Interpret data in picture Interpret picture graphs with Interpret/ Interpret picture and bar Interpret tally charts, bar Analyze Data pictographs. graphs, tally charts, and bar scales. Analyze Data graphs with scales. graphs, picture graphs, tables, graphs. and line graphs.

Read bar graphs with scales. Analyze data in a double bar graph.

Solve problems involving data. Solve real-world problems using Use frequency tables, bar Find the mean (average), picture graphs. graphs, picture graphs, and median, mode, and range of a line plots to solve real-world data set. problems. Probability Probability Outcomes Outcomes Decide whether an outcome Determine experimental is certain, more likely, equally probability of an outcome. likely, less likely, or impossible.

Expressing Expressing Express the probability of an Compare the results of an Probability Probability event as a fraction. experiment with theoretical probability.

Find all possible combinations by listing, making a tree diagram, and multiplying. Problem Solving Problem Solving Build Skills Build skills in addition and Build skills in addition, Build skills in addition, Build Skills Build skills in addition, Build skills in multiplication, Build skills in multiplication; Through subtraction through problem subtraction, and measurement subtraction, multiplication, Through subtraction, multiplication, division, fraction concepts, division; fraction concepts, Problem solving. through problem solving. division, and measurement Problem division, and measurement data analysis, and decimals, ratios, and percents; through problem solving. through problem solving. measurement through problem data analysis; and measurement Solving Solving solving. through problem solving.

28 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Data Analysis (continued) Data Analysis (continued) Collect and Organize data for a picture Collect and organize data in Collect and organize data in Collect and Organize Data graph. different ways. different ways. Organize Data

Represent Data Represent data in pictographs. Represent measurements and Represent data in picture Represent Data data in picture graphs, tally graphs. charts, and bar graphs.

Read bar graphs with scales. Analyze data in a double bar graph.

Expressing Expressing Express the probability of an Compare the results of an Probability Probability event as a fraction. experiment with theoretical probability.

29 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Problem Solving (continued) Problem Solving (continued) Solve Solve real-world problems Solve real-world problems Solve real-world problems Solve Solve real-world problems Solve real-world problems Solve real-world problems Real-World involving addition and involving addition and involving addition, subtraction, Real-World involving addition, subtraction, involving multiplication, involving multiplication; Problems subtraction. subtraction. multiplication, division, and Problems multiplication, division, and division, fraction concepts, division; concepts with fractions, measurement. measurement. data analysis, and decimals, ratios, and percents; measurement. data analysis; and measurement. Use Apply problem solving Apply problem solving Use Apply problem solving Use appropriate strategies to Use appropriate strategies to Appropriate strategies in Put on Your strategies in Put on Your Appropriate strategies in Put on Your solve real-world problems. solve real-world problems. Strategies and Thinking Cap! and Problem Thinking Cap! and Problem Strategies and Thinking Cap! and Problem Solving activities. Solving activities. Solving activities. Thinking Skills Thinking Skills to Solve to Solve Problems Problems Apply and Solve real-world problems. Apply and explain problem Apply and explain problem Apply and Apply and explain problem Apply and explain problem Apply and explain problem Explain solving processes in Put on solving processes in Put on Explain solving processes in Put on solving processes in Put on solving processes in Put on Problem Your Thinking Cap! and other Your Thinking Cap! and other Problem Your Thinking Cap! and other Your Thinking Cap! and other Your Thinking Cap! and other activities. activities. activities. activities. activities. Solving Solving Explore Use models to explain Explore concepts more deeply Explore concepts more deeply Explore Explore concepts more deeply Explore concepts more deeply Explore concepts more deeply Concepts reasoning. and justify reasoning in and justify reasoning in Concepts and justify reasoning in and justify reasoning in and justify reasoning in Let’s Let’s Explore and Hands-On Let’s Explore and Hands-On Let’s Explore and Hands-On Let’s Explore and Hands-On Explore and Hands-On activities. activities. activities. activities. activities. Apply Thinking Skills, Put Apply Thinking Skills, Put on Apply Thinking Skills, Put Apply Thinking Skills, Put Apply Thinking Skills, Put on on Your Thinking Cap!, Your Thinking Cap!, Challenging on Your Thinking Cap!, on Your Thinking Cap!, Your Thinking Cap!, Challenging Challenging Practice, and Practice, and Problem Solving Challenging Practice, and Challenging Practice, and Practice, and Problem Solving Problem Solving activities. activities. Problem Solving activities. Problem Solving activities. activities. Investigate Investigate ideas with two- Further investigate Further investigate Investigate Further investigate Further investigate Further investigate Mathematical dimensional shapes. mathematical ideas by mathematical ideas by Mathematical mathematical ideas by mathematical ideas by mathematical ideas by Ideas completing critical thinking completing critical thinking Ideas completing critical thinking completing critical thinking completing critical thinking skills skills activities. skills activities. skills activities. skills activities. activities. Identify, Demonstrate that only a few Explore transitivity by Demonstrate the inverse Identify, Demonstrate that figures Show that some figures can be Apply the idea that the sum of Demonstrate, big things fit into small spaces comparing lengths and weights relationship between the Demonstrate, and their flip, slide, and turn turned and not change shape angles on a straight line is 180°. and Explain and many small things fit into of three different objects. size of a unit and the number of and Explain images are congruent. or size (rotational symmetry). big spaces. units. Mathematical Mathematical Proof Proof Apply the idea that the sum of angles at a point is 360°.

Describe, sort, and classify two- Identify and describe attributes Identify, describe, sort, and Identify pairs of shapes that Use properties of squares and Explain the relationships among and three-dimensional shapes. and properties of two- and classify two- and three- show a flip, slide, and turn. rectangles to solve problems. area formulas of different three-dimensional shapes. dimensional shapes. polygons.

30 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Problem Solving (continued) Problem Solving (continued) Solve Solve real-world problems Solve real-world problems Solve real-world problems Solve Solve real-world problems Solve real-world problems Solve real-world problems Real-World involving addition and involving addition and involving addition, subtraction, Real-World involving addition, subtraction, involving multiplication, involving multiplication; Problems subtraction. subtraction. multiplication, division, and Problems multiplication, division, and division, fraction concepts, division; concepts with fractions, measurement. measurement. data analysis, and decimals, ratios, and percents; measurement. data analysis; and measurement. Use Apply problem solving Apply problem solving Use Apply problem solving Use appropriate strategies to Use appropriate strategies to Appropriate strategies in Put on Your strategies in Put on Your Appropriate strategies in Put on Your solve real-world problems. solve real-world problems. Strategies and Thinking Cap! and Problem Thinking Cap! and Problem Strategies and Thinking Cap! and Problem Solving activities. Solving activities. Solving activities. Thinking Skills Thinking Skills to Solve to Solve Problems Problems Apply and Solve real-world problems. Apply and explain problem Apply and explain problem Apply and Apply and explain problem Apply and explain problem Apply and explain problem Explain solving processes in Put on solving processes in Put on Explain solving processes in Put on solving processes in Put on solving processes in Put on Problem Your Thinking Cap! and other Your Thinking Cap! and other Problem Your Thinking Cap! and other Your Thinking Cap! and other Your Thinking Cap! and other activities. activities. activities. activities. activities. Solving Solving Explore Use models to explain Explore concepts more deeply Explore concepts more deeply Explore Explore concepts more deeply Explore concepts more deeply Explore concepts more deeply Concepts reasoning. and justify reasoning in and justify reasoning in Concepts and justify reasoning in and justify reasoning in and justify reasoning in Let’s Let’s Explore and Hands-On Let’s Explore and Hands-On Let’s Explore and Hands-On Let’s Explore and Hands-On Explore and Hands-On activities. activities. activities. activities. activities. Apply Thinking Skills, Put Apply Thinking Skills, Put on Apply Thinking Skills, Put Apply Thinking Skills, Put Apply Thinking Skills, Put on on Your Thinking Cap!, Your Thinking Cap!, Challenging on Your Thinking Cap!, on Your Thinking Cap!, Your Thinking Cap!, Challenging Challenging Practice, and Practice, and Problem Solving Challenging Practice, and Challenging Practice, and Practice, and Problem Solving Problem Solving activities. activities. Problem Solving activities. Problem Solving activities. activities. Investigate Investigate ideas with two- Further investigate Further investigate Investigate Further investigate Further investigate Further investigate Mathematical dimensional shapes. mathematical ideas by mathematical ideas by Mathematical mathematical ideas by mathematical ideas by mathematical ideas by Ideas completing critical thinking completing critical thinking Ideas completing critical thinking completing critical thinking completing critical thinking skills skills activities. skills activities. skills activities. skills activities. activities. Identify, Demonstrate that only a few Explore transitivity by Demonstrate the inverse Identify, Demonstrate that figures Show that some figures can be Apply the idea that the sum of Demonstrate, big things fit into small spaces comparing lengths and weights relationship between the Demonstrate, and their flip, slide, and turn turned and not change shape angles on a straight line is 180°. and Explain and many small things fit into of three different objects. size of a unit and the number of and Explain images are congruent. or size (rotational symmetry). big spaces. units. Mathematical Mathematical Proof Proof Apply the idea that the sum of angles at a point is 360°.

31 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Problem Solving (continued) Problem Solving (continued) Identify, Interpret data in tally charts and Interpret picture graphs, tally Interpret picture graphs Identify, Interpret bar graphs with Analyze a data set by finding Compare the results of an Demonstrate, pictographs. charts, and bar graphs. with scales. Demonstrate, scales. its mean, median, mode, and experiment to validate the use of and Explain and Explain range. theoretical probability. Mathematical Mathematical Proof Proof (continued) (continued) Identify and extend repeating Identify and extend growing Identify rules for number Create and analyze Identify, describe, and extend Identify, describe, and extend shape patterns. number patterns and repeating patterns. multiplication and division numeric and non-numeric numeric patterns involving shape patterns. patterns. patterns. all operations. Use a Variety of Sort and classify using Recognize shapes from Identify surfaces that slide, Use a Variety of Model, define, and explain Use properties of squares and Explore the relationship Reasoning Skills attributes. different perspectives. stack, and roll. Reasoning Skills properties of multiplication. rectangles to solve problems among lists, tree diagrams, about area and perimeter. and multiplication to calculate combinations. Identify similarities and Use the Commutative and Explore the inverse relationship Explore the inverse relationship Use properties of multiplication differences. Associative properties, and between addition and between multiplication and (including 10s and 1s to solve two-digit subtraction. division. the Distributive Property) in addition and subtraction estimation and mental math. problems. Use estimation to check Use estimation to check reasonableness. reasonableness (whole- number addition, subtraction, multiplication and division). Communication Communication Consolidate Consolidate thinking in Present mathematical thinking Present mathematical thinking Consolidate Present mathematical thinking Present mathematical thinking Present mathematical thinking Mathematical independent activities. through Math Journal activities. through Math Journal activities. Mathematical through Math Journal activities. through Math Journal activities. through Math Journal activities. Thinking Thinking Communicate Discuss mathematical ideas Discuss mathematical ideas in Discuss mathematical ideas in Communicate Discuss mathematical ideas in Discuss mathematical ideas in Discuss mathematical ideas with Peers, in paired and small-group Let’s Explore activities. Let’s Explore activities. with Peers, Let’s Explore activities. Let’s Explore activities. in Let’s Explore activities. Teachers, and activities. Teachers, and Others Others Work together in pairs or Work together in pairs or groups Work together in pairs or Work together in pairs or Work together in pairs or groups groups in Let’s Explore, Games, in Let’s Explore, Games, and groups in Let’s Explore, Games, groups in Let’s Explore, Games, in Let’s Explore, Games, and and other activities. other activities. and other activities. and other activities. other activities. Share Share mathematical ideas Share mathematical ideas with Share mathematical ideas with Share Share mathematical ideas with Share mathematical ideas with Share mathematical ideas with Mathematical in paired and small-group others during Let’s Explore and others during Let’s Explore and Mathematical others during Let’s Explore and others during Let’s Explore and others during Let’s Explore and Thinking activities. Hands-On activities. Hands-On activities. Thinking Hands-On activities. Hands-On activities. Hands-On activities.

32 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Problem Solving (continued) Problem Solving (continued) Identify, Interpret data in tally charts and Interpret picture graphs, tally Interpret picture graphs Identify, Interpret bar graphs with Analyze a data set by finding Compare the results of an Demonstrate, pictographs. charts, and bar graphs. with scales. Demonstrate, scales. its mean, median, mode, and experiment to validate the use of and Explain and Explain range. theoretical probability. Mathematical Mathematical Proof Proof (continued) (continued) Identify and extend repeating Identify and extend growing Identify rules for number Create and analyze Identify, describe, and extend Identify, describe, and extend shape patterns. number patterns and repeating patterns. multiplication and division numeric and non-numeric numeric patterns involving shape patterns. patterns. patterns. all operations. Use a Variety of Sort and classify using Recognize shapes from Identify surfaces that slide, Use a Variety of Model, define, and explain Use properties of squares and Explore the relationship Reasoning Skills attributes. different perspectives. stack, and roll. Reasoning Skills properties of multiplication. rectangles to solve problems among lists, tree diagrams, about area and perimeter. and multiplication to calculate combinations. Identify similarities and Use the Commutative and Explore the inverse relationship Explore the inverse relationship Use properties of multiplication differences. Associative properties, and between addition and between multiplication and (including 10s and 1s to solve two-digit subtraction. division. the Distributive Property) in addition and subtraction estimation and mental math. problems. Use estimation to check Use estimation to check reasonableness. reasonableness (whole- number addition, subtraction, multiplication and division). Communication Communication Consolidate Consolidate thinking in Present mathematical thinking Present mathematical thinking Consolidate Present mathematical thinking Present mathematical thinking Present mathematical thinking Mathematical independent activities. through Math Journal activities. through Math Journal activities. Mathematical through Math Journal activities. through Math Journal activities. through Math Journal activities. Thinking Thinking Communicate Discuss mathematical ideas Discuss mathematical ideas in Discuss mathematical ideas in Communicate Discuss mathematical ideas in Discuss mathematical ideas in Discuss mathematical ideas with Peers, in paired and small-group Let’s Explore activities. Let’s Explore activities. with Peers, Let’s Explore activities. Let’s Explore activities. in Let’s Explore activities. Teachers, and activities. Teachers, and Others Others Work together in pairs or Work together in pairs or groups Work together in pairs or Work together in pairs or Work together in pairs or groups groups in Let’s Explore, Games, in Let’s Explore, Games, and groups in Let’s Explore, Games, groups in Let’s Explore, Games, in Let’s Explore, Games, and and other activities. other activities. and other activities. and other activities. other activities. Share Share mathematical ideas Share mathematical ideas with Share mathematical ideas with Share Share mathematical ideas with Share mathematical ideas with Share mathematical ideas with Mathematical in paired and small-group others during Let’s Explore and others during Let’s Explore and Mathematical others during Let’s Explore and others during Let’s Explore and others during Let’s Explore and Thinking activities. Hands-On activities. Hands-On activities. Thinking Hands-On activities. Hands-On activities. Hands-On activities.

33 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Communication (continued) Communication (continued) Express Express ideas in paired and Express ideas in Math Journal Express ideas in Math Journal Express Express ideas in Math Journal Express ideas in Math Journal Express ideas in Math Journal Mathematical small group activities. activities, using lesson activities, using lesson Mathematical activities, using lesson activities, using lesson activities, using lesson Ideas vocabulary. vocabulary. Ideas vocabulary. vocabulary. vocabulary. Use chapter and lesson Use chapter and lesson Use chapter and lesson Use chapter and lesson Use chapter and lesson vocabulary correctly. vocabulary correctly. vocabulary correctly. vocabulary correctly. vocabulary correctly.

Connections Connections Recognize Con- Understand the connection Understand the relationship Examine and apply the inverse Recognize Con- Apply the inverse relationship Demonstrate that decimal Relate fractions and division. nections in between quantities and written between counting and addition relationship between addition nections in between multiplication and notation is an extension of the Mathematical numerals. and subtraction. and subtraction. Mathematical division. base-ten system. Ideas Ideas Understand the relationships Connect geometric concepts Understand that the size of a Examine the relationship Understand the connection among the numbers in fact with unit fractions. fractional part is relative to the between fractions and among fractions, decimals, families. size of the whole. decimals. ratios, and percents as ways to represent parts of a whole.

Connect addition and Connect subtraction and Connect the units of customary Make connections among Examine the relationships multiplication (repeated division (repeated subtraction). capacity to one another. multiplication, division, factors, between three-dimensional addition). and multiples. figures and the two-dimensional figures that form them.

Recognize and apply different Recognize and apply different Understand the relationships Convert among mixed Relate fractions and division. strategies for adding and strategies for multiplication and between the numbers in numbers and improper subtracting division facts. multiplication-division fact fractions. one- and two-digit numbers. families. Understand Explore relationships among Learn how place value concepts Understand how patterns can Understand Understand the meanings Describe number relationships Explain the relationships among How Concepts counting, ordering, and ordinal apply to regrouping in addition be described using numbers, How Concepts and uses of fractions including in context. area formulas of different Build on One numbers. and subtraction. operations, and data displays. Build on One fraction of a set. polygons. Another Another Connect equivalent fractions Connect equivalent fractions, and decimals. decimals, and percents.

Recognize the relationship Use addition, subtraction, Make connections among between bar models, number multiplication, and division to the greatest common factor, sentences, and number construct and analyze graphs, least common multiple, and patterns. frequency tables, and line operations with fractions. plots.

34 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Communication (continued) Communication (continued) Express Express ideas in paired and Express ideas in Math Journal Express ideas in Math Journal Express Express ideas in Math Journal Express ideas in Math Journal Express ideas in Math Journal Mathematical small group activities. activities, using lesson activities, using lesson Mathematical activities, using lesson activities, using lesson activities, using lesson Ideas vocabulary. vocabulary. Ideas vocabulary. vocabulary. vocabulary. Use chapter and lesson Use chapter and lesson Use chapter and lesson Use chapter and lesson Use chapter and lesson vocabulary correctly. vocabulary correctly. vocabulary correctly. vocabulary correctly. vocabulary correctly.

35 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Connections (continued) Connections (continued) Solve Solve real-world problems Solve real-world problems Solve real-world problems Solve Solve real-world problems Solve real-world problems Solve real-world problems Real-World involving more and less. involving addition, subtraction, involving addition, subtraction, Real-World involving addition, subtraction, involving multiplication, involving multiplication; Problems in and measurement. multiplication, division, Problems in multiplication, division, and division, fraction concepts, division; fraction, decimal, ratio, measurement, and data measurement. data analysis, and and percent concepts; data Contexts analysis. Contexts measurement. analysis; and measurement. Outside of Outside of Mathematics Mathematics Solve real-world problems Compare experimental results related to money. and theoretical probability.

Representation Representation Use Use concrete models to create Use concrete and pictorial Use concrete and pictorial Use Use place value models to Represent numbers to 100,000 Explore negative numbers in Representations a set with a given number of models to create a set with a models to create a set with Representations read, write, and represent in various contexts. context. to Model, objects (up to 20). given number of objects (up to a given number of objects (up to to Model, numbers to 10,000. 100). 1,000). Organize, and Organize, and Record Record Use numbers and numerals to Represent numbers to 100 on a Represent numbers to 1,000 on Represent numbers in different Express numbers to 100,000 in Express numbers to 10,000,000 represent quantities up to 20. number line. a number line. equivalent forms. standard, expanded, and word in various forms. forms.

Use picture cards to Use number bonds to represent communicate understanding numbers. of comparisons (bigger and smaller). Understand the meaning of the Understand equality and Use symbolic notation (< and >) Use the dollar sign and decimal Model decimals to tenths and Find equivalent ratios. = sign in number sentences. inequality. to compare numbers. point in money amounts. hundredths.

Model addition and subtraction Use the +, −, and = symbols to Use bar models to represent Solve addition and subtraction Write addition and subtraction Explore the use of letters as stories with addition and represent real-world addition addition and subtraction problems with greater number sentences for real- variables in expressions and subtraction number sentences. and subtraction situations. situations. numbers by using a bar model world problems with fractions inequalities. and decimals. Represent addition and Represent numerical data Represent numerical data using Represent multiplication and Use models to show Convert fractions and decimals subtraction stories. using picture graphs, tally picture graphs with scales, tally division in different ways. relationships between to percents. charts, and bar graphs. charts, and bar graphs. improper fractions and mixed numbers. Represent sharing equally and Use the ×, ÷, and = symbols to Use a variety of representations Apply understanding of models Represent combinations making equal groups. represent multiplication and for multiplication and division, for multiplication and division. with lists, tree diagrams, and division situations. such as arrays, area models, multiplication. number lines, grouping, and sharing.

36 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Connections (continued) Connections (continued) Solve Solve real-world problems Solve real-world problems Solve real-world problems Solve Solve real-world problems Solve real-world problems Solve real-world problems Real-World involving more and less. involving addition, subtraction, involving addition, subtraction, Real-World involving addition, subtraction, involving multiplication, involving multiplication; Problems in and measurement. multiplication, division, Problems in multiplication, division, and division, fraction concepts, division; fraction, decimal, ratio, measurement, and data measurement. data analysis, and and percent concepts; data Contexts analysis. Contexts measurement. analysis; and measurement. Outside of Outside of Mathematics Mathematics Solve real-world problems Compare experimental results related to money. and theoretical probability.

37 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Representation (continued) Representation (continued) Use Represent multiplication with Use Determine the missing parts Write addition and subtraction Write and solve equations. Representations skip counting, dot paper arrays, Representations (quantities or symbols) in number sentences for real- to Model, and bar models. to Model, number sentences. world problems with fractions and decimals. Organize, and Organize, and Record Record (continued) (continued) Represent division as repeated subtraction sentences.

Describe and extend shape Identify, describe, and extend Describe, extend, and create Create and analyze patterns. two- and three-dimensional two-dimensional shape multiplication and division shape patterns. patterns. patterns. Describe a rule for sorting Identify a rule for sorting objects. objects.

Identify and extend growing Identify rules for number Identify a rule for number and Use a rule to describe a Find rules to complete number and repeating patterns. patterns. counting patterns. sequence of numbers or patterns. objects.

Select and Represent quantities with Use number bonds to represent Use place value models to create Select and Use a variety of models to Translate between equivalent Translate between fractions and Apply objects, number cubes, and number combinations. equivalent representations of Apply represent fractions and improper fractions and mixed percents. Representations numerals. numbers. Representations equivalent fractions. numbers. to Model to Model Problems Problems Use a variety of concrete, Use a variety of concrete, Use a variety of concrete, Use a variety of models for Select the most useful form of pictorial, and symbolic models pictorial, and symbolic models pictorial, and symbolic models multi-digit multiplication and the quotient. for addition and subtraction. for addition, subtraction, for multi-digit addition, division of whole numbers. multiplication, and division. subtraction, multiplication, and division. Represent multiplication with Use customary units (including Use a variety of models for Use a net to find the surface area skip counting and arrays. fractions) to measure length, addition and subtraction of of a prism. weight, and capacity. fractions and decimals. Interpret Show understanding of big, Measure and compare lengths Use metric and customary units Interpret Use referents to estimate Measure perimeter and area in Measure volume of a rectangular Phenomena middle-sized, small, and same and weights using non- to measure length, volume Phenomena length, capacity, and weight. customary and metric units. prism. through size. standard units. (capacity), weight, and mass. through Representations Representations Describe and compare objects Use positional words to by position. describe location.

38 Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Representation (continued) Representation (continued) Use Represent multiplication with Use Determine the missing parts Write addition and subtraction Write and solve equations. Representations skip counting, dot paper arrays, Representations (quantities or symbols) in number sentences for real- to Model, and bar models. to Model, number sentences. world problems with fractions and decimals. Organize, and Organize, and Record Record (continued) (continued) Represent division