 ## 5

### Counting and Cardinality

Know number names and the count sequence.
1 Count to 100 by ones and by tens.
2 Count to 100 by ones beginning from any given number (instead of beginning at 1).
3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
Count to tell the number of objects.
4 Understand the relationship between numbers and quantities up to 20; connect counting to cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
Understand the concept of ordinal numbers (first through tenth) to describe the relative position and magnitude of whole numbers.
5
a. Answer counting questions using as many as 20 objects arranged in a line, a rectangular array, and a circle. Answer counting questions using as many as 10 objects in a scattered configuration.
b. Given a number from 1â€“20, count out that many objects.
Compare numbers.
6 Identify whether the number of objects in one group is greater than (more than), less than (fewer than), or equal to (the same as) the number of objects in another group.
7 Compare two numbers between 1 and 10 presented as written numerals.

### Operations and Algebraic Thinking

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
1 Represent addition and subtraction using objects, fingers, pennies, drawings, sounds, acting out situations, verbal explanations, expressions, equations, or other strategies.
2a Add and subtract within 10.
2b Solve addition and subtraction word problems within 10.
3 Decompose numbers less than or equal to 10 into pairs in more than one way. Record each decomposition with a drawing or equation.
4 Find the number that makes 10 when given a number from 1 to 9. Record the answer with a drawing or equation.
5 Fluently add and subtract within 5.
6 Duplicate, extend, and create simple patterns using concrete objects.

### Number and Operations in Base Ten

Work with numbers 11â€“19 to gain foundations for place value.
1 Compose and decompose the numbers from 11 to 19 into ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

### Measurement and Data

Describe and compare measurable attributes.
1 Describe measurable attributes of an object(s), such as length or weight, using appropriate vocabulary.
2 Directly compare two objects with a common measurable attribute and describe the difference.
Classify objects and count the number of objects in each category.
3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
4 Explore coins (pennies, nickels, dimes, and quarters) and begin identifying pennies and dimes.

### Geometry

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
2 Name shapes regardless of their orientation or overall size.
3 Understand the difference between two-dimensional (lying in a plane, â€œflatâ€) and three- dimensional (â€œsolidâ€) shapes.
Analyze, compare, sort, and compose shapes.
4 Analyze, compare, and sort two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts, and other attributes.
5 Model objects in their environment by building and/or drawing shapes.
6 Compose larger shapes from simple shapes.

### Operations and Algebraic Thinking

Represent and solve problems involving addition and subtraction.
1 Use addition and subtraction within 20 to solve one step word problems involving situations of adding to, taking from, putting together, taking apart, and/or comparing, with unknowns in all positions.
2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.
Understand and apply properties of operations and the relationship between addition and subtraction.
3 Apply properties of operations as strategies to add and subtract
4 Understand subtraction as an unknown-addend problem within 20.
5 Relate counting to addition and subtraction.
6a

Add and subtract within 20. Use strategies such as:

• counting on;
• making ten;
• decomposing a number leading to a ten;
• using the relationship between addition and subtraction; and
• creating equivalent but easier or known sums.
6b

Fluently add and subtract within 10.

Work with addition and subtraction equations.
7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
8 Determine the unknown whole number in an addition or subtraction equation with the unknown in all positions.

### Number & Operations in Base Ten

Extend the counting sequence.
1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Understand place value.
2 Understand that the two digits of a two-digit number represent amounts of tens and ones.
a. Understand 10 can be thought of as a bundle of ten ones, called a "ten".
b. Understand the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. Understand the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
Use place value understanding and properties of operations to add and subtract.
4

• a two-digit number and a one-digit number,
• a two-digit number and a multiple of 10.

Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Understand that in adding two-digit numbers, one adds tens and tens, ones and ones, and sometimes it is necessary to compose a ten.

Relate the strategy to a written representation and explain the reasoning used.

5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6

Subtract multiples of 10 from multiples of 10 in the range 10-90 using

• concrete models or drawings, and
• strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Relate the strategy used to a written representation and explain the reasoning.

### Measurement & Data

Measure lengths indirectly and by iterating length units.
1 Order three objects by length; compare the lengths of two objects indirectly by using a third object.
2 Measure the length of an object using same-size â€œlength unitsâ€ placed end to end with no gaps or overlaps. Express the length of an object as a whole number of â€œlength units.â€
Tell and write time and money.
3a Tell and write time in hours and half-hours using analog and digital clocks. Develop an understanding of common terms, such as, but not limited to, o’clock and half past.
3b Recognize and identify coins (penny, nickel, dime, and quarter) and their value and use the cent symbol (¢) appropriately.
3c Count a mixed collection of dimes and pennies and determine the cent value (total not to exceed 100 cents).
C. Represent and interpret data.
4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

### Geometry

Reason with shapes and their attributes.
1 Distinguish between defining attributes versus non-defining attributes for a wide variety of shapes. Build and/or draw shapes to possess defining attributes.
2

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

### Operations and Algebraic Thinking

Represent and solve problems involving addition and subtraction.
1a

Use addition and subtraction within 100 to solve one-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

1b

Use addition and subtraction within 100 to develop an understanding of solving two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

2a

Fluently add and subtract within 20 using mental strategies. Strategies could include:

• counting on;
• making ten;
• decomposing a number leading to a ten;
• using the relationship between addition and subtraction; and
• creating equivalent but easier or known sums.
2b

Know from memory all sums within 20 of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.
3a

Determine whether a group of objects (up to 20) has an odd or even number of members.

3b

Write an equation to express an even number as a sum of two equal addends.

4

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as a sum of equal addends.

### Number and Operations in Base Ten

Understand Place Value
1

Understand that the digits of a three-digit number represent amounts of hundreds, tens, and ones.

a. Understand 100 can be thought of as a bundle of ten tens, called a â€œhundred.â€
b. Understand the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2

Count within 1000; skip-count by 5s, 10s, and 100s.

3

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

4

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Use place value understanding and properties of operations to add and subtract.
5

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

6

Add up to four two-digit numbers using strategies based on place value and properties of operations.

7a

Add and subtract within 1000, using

• concrete models or drawings, and
• strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Relate the strategy to a written representation.

7b

Understand that in adding or subtracting up to three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and sometimes it is necessary to compose or decompose tens or hundreds.

8

Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

9

Explain why addition and subtraction strategies work, using place value and the properties of operations.

### Measurement and Data

Measure and estimate lengths in standard units.
1

Measure the length of an object to the nearest whole by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

2

Measure the length of an object twice, using different “length units” for the two measurements; describe how the two measurements relate to the size of the unit chosen.

3

Estimate lengths using units of inches, feet, centimeters, and meters.

4

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

Relate addition and subtraction to length.
5

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.

6

Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line.

Work with time and money.
7

Tell and write time from analog and digital clocks in five minute increments, using a.m. and p.m. Develop an understanding of common terms, such as, but not limited to, quarter past, half past, and quarter to.

8a

Count a mixed collection of coins whose sum is less than or equal to one dollar.

8b

Solve real world and mathematical problems within one dollar involving quarters, dimes, nickels, and pennies, using the Â¢ (cent) symbol appropriately.

Represent and interpret data.
9

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Present the measurement data in a line plot, where the horizontal scale is marked off in whole-number units.

10

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a picture graph or a bar graph.

### Geometry

Reason with shapes and their attributes.
1

Classify two-dimensional figures as polygons or non-polygons.

2

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

3

Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the words halves, thirds, half of, a third of, etc. Describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

### Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.
1

Interpret products of whole numbers.

2

Interpret whole-number quotients of whole numbers.

3

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

4

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

Understand properties of multiplication and the relationship between multiplication and division.
5

Apply properties of operations as strategies to multiply and divide.

6

Understand division as an unknown-factor problem.

Multiply and divide within 100
7a

Fluently solve single-digit multiplication and related divisions, using strategies such as the relationship between multiplication and division or properties of operations.

7b

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8a

Solve two-step word problems posed with whole numbers and having whole-number answers using the four operations.

a. Represent these problems using equations or expressions with a letter standing for the unknown quantity.

8b

Solve two-step word problems posed with whole numbers and having whole-number answers using the four operations.

b. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

9

Identify and extend arithmetic patterns (including patterns in the addition table or multiplication table).

### Number and Operations in Base Ten

Use place value understanding and properties of operations to perform multi-digit arithmetic.
1

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

2

Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3

Multiply one-digit whole numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties of operations.

4a

Understand that the digits of a four-digit number represent amounts of thousands, hundreds, tens, and ones.

4b

Read and write four-digit numbers using base-ten numerals, number names, and expanded form.

### Number and Operations--Fractions

Develop understanding of fractions as numbers.
1

Understand a unit fraction, 1/b, is the quantity formed by 1 part when a whole is partitioned into b equal parts. Understand a fraction a/b as the quantity formed by a parts of size 1/b.

2

Understand a fraction as a number on the number line; represent fractions on a number line.

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons rely on the two fractions referring to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

### Measurement and Data

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
1

Tell and write time to the nearest minute and measure time intervals in minutes. Solve one-step word problems involving addition and subtraction of time intervals in minutes.

2a

Measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (l).

2b

Add, subtract, multiply, or divide to solve one-step word problems involving masses or liquid volumes that are given in the same units.

Represent and interpret data
3

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step â€œhow many moreâ€ and â€œhow many lessâ€ problems using information presented in a scaled picture graph or a scaled bar graph.

4

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate unitsâ€”whole numbers, halves, or quarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
5

Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. Recognize a square with side length 1 unit, called â€œa unit square,â€ is said to have â€œone square unitâ€ of area, and can be used to measure area.
b. Recognize a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
6

Measure areas by counting unit squares.

7

Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side length a and side length b + c is the sum of a Ã— b and a Ã— c. Use area models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of figures composed of non-overlapping rectangles, and apply this technique to solve real world problems.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
8a

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths or finding one unknown side length given the perimeter and other side lengths.

8b

Identify rectangles with the same perimeter and different areas or with the same area and different perimeters.

### Geometry

Reason with shapes and their attributes
1

Recognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons, and hexagons). Identify shapes that do not belong to one of the given subcategories.

2

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

### Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems.
1

Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations.

2

Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison. Use drawings and equations with a symbol for the unknown number to represent the problem.

3a

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.

a. Represent these problems using equations or expressions with a letter standing for the unknown quantity.

3b

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.

b. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Gain familiarity with factors and multiples.
4

Find all factor pairs for a whole number in the range 1â€“100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1â€“100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Generate and analyze patterns.
5

Generate a number or shape pattern that follows a given rule. Identify and informally explain apparent features of the pattern that were not explicit in the rule itself.

### Number and Operations in Base Ten

Generalize place value understanding for multi-digit whole numbers.
1

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

2a

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.

2b

Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

3

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

Use place value understanding and properties of operations to perform multi-digit arithmetic.
4

Fluently add and subtract multi-digit whole numbers using a standard algorithm.

5

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

6

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

### Number and Operations--Fractions

Extend understanding of fraction equivalence and ordering.
1

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

2

Compare two fractions with different numerators and different denominators.

Recognize that comparisons are valid only when the two fractions refer to the same whole.

Record the results of comparisons with symbols >, =, or <, and justify the conclusions.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3

Understand a fraction a/b with a > 1 as a sum of fractions 1/b

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions.
c. Add and subtract mixed numbers with like denominators.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
4

Apply and extend previous understandings of multiplication to multiply a whole number by a fraction.

a. Understand a fraction a/b as a multiple of 1/b.
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a whole number by a fraction.
c. Solve word problems involving multiplication of a whole number by a fraction.
Understand decimal notation for fractions, and compare decimal fractions.
5

Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.

6

Use decimal notation for fractions with denominators 10 or 100.

7

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

### Measurement and Data

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1

Know relative sizes of measurement units: ft., in.; km, m, cm

Know the conversion factor and use it to convert measurements in a larger unit in terms of a smaller unit: ft., in.; km, m, cm; hr., min., sec.

Given the conversion factor, convert all other measurements within a single system of measurement from a larger unit to a smaller unit.

Record measurement equivalents in a two-column table.

2a

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money.

a. Solve problems involving fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.

2b

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money.

b. Represent measurement quantities using diagrams that feature a measurement scale, such as number lines.

3

Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

Represent and interpret data.
4

Make a line plot to display a data set of measurements in fractions of a unit ô‰€12,14,18ô‰. Solve problems involving addition and subtraction of fractions by using information presented in line plots.

Geometric measurement: understand concepts of angle and measure angles.
5

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.

a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a â€œone-degree angle,â€ and can be used to measure angles.
b. Recognize an angle that turns through n one-degree angles is said to have an angle measure of n degrees.
6

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

7

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.

### Geometry

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
1

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

2a

Identify and name triangles based on angle size (right, obtuse, acute).

2b

Identify and name all quadrilaterals with 2 pairs of parallel sides as parallelograms.

2c

Identify and name all quadrilaterals with four right angles as rectangles.

3

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

### Operations and Algebraic Thinking

Write and interpret numerical expressions
1

Apply the order of operations to evaluate numerical expressions.

2

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

B. Analyze patterns and relationships
3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

### Number and Operations in Base Ten

Understand the place value system.
1

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

2

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

3

Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4

Use place value understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with decimals to hundredths.
5

Fluently multiply multi-digit whole numbers using a standard algorithm.

6

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

7

Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations:

• add and subtract decimals to hundredths;
• multiply and divide decimals to hundredths.

Relate the strategy to a written method and explain the reasoning used.

### Number and Operations--Fractions

Use equivalent fractions as a strategy to add and subtract fractions.
1

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

2

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators.

Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3

Interpret a fraction as division of the numerator by the denominator (a/b = a Ã· b).

Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction.

a. Interpret the product (a/b) Ã— q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a Ã— q Ã· b.
b. Find the area of a rectangle with fractional side lengths by tiling it with rectangles of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5

Interpret multiplication as scaling (resizing).

a. Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case). Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. Relate the principle of fraction equivalence
a/b = a/b x n/n to the effect of multiplying a/b by 1.
6

Solve real world problems involving multiplication of fractions and mixed numbers.

7

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
b. Interpret division of a whole number by a unit fraction, and compute such quotients.
c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

### Measurement and Data

Convert like measurement units within a given measurement system.
1

Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Represent and interpret data.
2

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. Recognize that a cube with side length 1 unit, called a â€œunit cube,â€ is said to have â€œone cubic unitâ€ of volume, and can be used to measure volume.
b. Recognize that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
4

Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.

5

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
b. Apply the formulas V = l Ã— w Ã— h and V = B Ã— h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

### Geometry

Graph points on the coordinate plane to solve real-world and mathematical problems.
1

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.

Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond.

2

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Classify two-dimensional figures into categories based on their properties.
3

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

4

Classify two-dimensional figures in a hierarchy based on properties.