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### 4.1 Number Sense, Properties, and Operations

- 4.1.1 The decimal number system to the hundredths place describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms
- 4.1.1.a Generalize place value understanding for multi-digit whole numbers.
- 4.1.1.a.i Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
- 4.1.1.a.ii Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.
- 4.1.1.a.iii Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- 4.1.1.a.iv Use place value understanding to round multi-digit whole numbers to any place.

- 4.1.1.b Use decimal notation to express fractions, and compare decimal fractions.
- 4.1.1.b.i 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.
- 4.1.1.b.ii Use decimal notation for fractions with denominators 10 or 100.
- 4.1.1.b.iii Compare two decimals to hundredths by reasoning about their size.

- 4.1.1.a Generalize place value understanding for multi-digit whole numbers.
- 4.1.2 Different models and representations can be used to compare fractional parts
- 4.1.2.a Use ideas of fraction equivalence and ordering to:
- 4.1.2.a.i Explain equivalence of fractions using drawings and models.
- 4.1.2.a.ii Use the principle of fraction equivalence to recognize and generate equivalent fractions.
- 4.1.2.a.iii Compare two fractions with different numerators and different denominators, and justify the conclusions.

- 4.1.2.b Build fractions from unit fractions by applying understandings of operations on whole numbers.
- 4.1.2.b.i Apply previous understandings of addition and subtraction to add and subtract fractions.
- 4.1.2.b.i.1 Compose and decompose fractions as sums and differences of fractions with the same denominator in more than one way and justify with visual models.
- 4.1.2.b.i.2 Add and subtract mixed numbers with like denominators.
- 4.1.2.b.i.3 Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

- 4.1.2.b.ii Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
- 4.1.2.b.ii.1 Express a fraction a/b as a multiple of 1/b.
- 4.1.2.b.ii.2 Use a visual fraction model to express a/b as a multiple of 1/b, and apply to multiplication of whole number by a fraction.
- 4.1.2.b.ii.3 Solve word problems involving multiplication of a fraction by a whole number.

- 4.1.2.b.i Apply previous understandings of addition and subtraction to add and subtract fractions.

- 4.1.2.a Use ideas of fraction equivalence and ordering to:
- 4.1.3 Formulate, represent, and use algorithms to compute with flexibility, accuracy, and efficiency
- 4.1.3.a Use place value understanding and properties of operations to perform multi-digit arithmetic.
- 4.1.3.a.i Fluently add and subtract multi-digit whole numbers using the standard algorithm.
- 4.1.3.a.ii 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.
- 4.1.3.a.iii 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.
- 4.1.3.a.iv Illustrate and explain multiplication and division calculation by using equations, rectangular arrays, and/or area models.

- 4.1.3.b Use the four operations with whole numbers to solve problems.
- 4.1.3.b.i Interpret a multiplication equation as a comparison.
- 4.1.3.b.ii Represent verbal statements of multiplicative comparisons as multiplication equations.
- 4.1.3.b.iii Multiply or divide to solve word problems involving multiplicative comparison.
- 4.1.3.b.iv 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.
- 4.1.3.b.v Represent multistep word problems with equations using a variable to represent the unknown quantity.
- 4.1.3.b.vi Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
- 4.1.3.b.vii Using the four operations analyze the relationship between choice and opportunity cost.

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

### 4.2 Patterns, Functions, and Algebraic Structures

- 4.2.1 Number patterns and relationships can be represented by symbols
- 4.2.1.a Generate and analyze patterns and identify apparent features of the pattern that were not explicit in the rule itself.
- 4.2.1.a.i Use number relationships to find the missing number in a sequence.
- 4.2.1.a.ii Use a symbol to represent and find an unknown quantity in a problem situation.
- 4.2.1.a.iii Complete input/output tables.
- 4.2.1.a.iv Find the unknown in simple equations.

- 4.2.1.b Apply concepts of squares, primes, composites, factors, and multiples to solve problems.
- 4.2.1.b.i Find all factor pairs for a whole number in the range 1-100.
- 4.2.1.b.ii Recognize that a whole number is a multiple of each of its factors.
- 4.2.1.b.iii Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number.
- 4.2.1.b.iv Determine whether a given whole number in the range 1-100 is prime or composite.

- 4.2.1.a Generate and analyze patterns and identify apparent features of the pattern that were not explicit in the rule itself.

### 4.3 Data Analysis, Statistics, and Probability

- 4.3.1 Visual displays are used to represent data
- 4.3.1.a Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
- 4.3.1.b Solve problems involving addition and subtraction of fractions by using information presented in line plots.

### 4.4 Shape, Dimension, and Geometric Relationships

- 4.4.1 Appropriate measurement tools, units, and systems are used to measure different attributes of objects and time
- 4.4.1.a Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
- 4.4.1.a.i Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec.
- 4.4.1.a.ii Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
- 4.4.1.a.iii Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
- 4.4.1.a.iv Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
- 4.4.1.a.v Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

- 4.4.1.b Use concepts of angle and measure angles.
- 4.4.1.b.i Describe angles as geometric shapes that are formed wherever two rays share a common endpoint, and explain concepts of angle measurement.
- 4.4.1.b.ii Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
- 4.4.1.b.iii Demonstrate that angle measure as additive.
- 4.4.1.b.iv Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.

- 4.4.1.a Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
- 4.4.2 Geometric figures in the plane and in space are described and analyzed by their attributes
- 4.4.2.a Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines.
- 4.4.2.b Identify points, line segments, angles, and perpendicular and parallel lines in two-dimensional figures.
- 4.4.2.c Classify and identify two-dimensional figures according to attributes of line relationships or angle size.
- 4.4.2.d Identify a line of symmetry for a two-dimensional figure.