Evidence Outcomes:

1. 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. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division. (CCSS: 4.NBT.A.1)

2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (CCSS: 4.NBT.A.2)

3. Use place value understanding to round multi-digit whole numbers to any place. (CCSS: 4.NBT.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Knowledge of place value patterns provides the foundation for using the four operations to solve problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core process students must understand are...

• In a multi-digit whole number, the value of a digit in one place represents 10 times what it represents in the place to its right and this pattern continues throughout the number

• Both the value and placement of digits determine the number’s overall value

In order to meet these essential understandings, students must know...

Academic Vocabulary: digit, standard form, unit form, word form, expanded form

• The name of each place in a number (ones, tens, hundreds, . . .)

• The meaning of each place in a number in terms of relative value

• How base ten models/manipulatives can represent and help with the understanding of multiplication and multiples of 10

• When it is helpful to use a rounded number instead of an exact number

In order to meet these essential understandings, students must be able to...

• Read and write multi-digit whole numbers using standard form, unit form, word form, and expanded form

• Compare the value (<, >, or =) of any multi-digit whole numbers

• Round multi-digit whole numbers to any place

• Use the structure of the base ten number system to read, write, compare, and round multi-digit numbers

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Fluently add and subtract multi-digit whole numbers using the standard algorithm. (CCSS: 4.NBT.B.4)

2. 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. (CCSS: 4.NBT.B.5)

3. 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. (CCSS: 4.NBT.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Examine and apply a variety of strategies to accurately and effectively solve problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core process students must understand are...

• Structure of place value to support the organization of mental and written multi-digit arithmetic strategies

• Relationships between and patterns/rules within operations

In order to meet these essential understandings, students must know...

Academic Vocabulary: algorithm, addend, sum, difference, factor, products, array, commutative, associative, distributive

• The standard algorithm for addition and subtraction

• Relationship between multiplication and division

• Multiple strategies that aid in understanding of multiplication, division, and place value

• Properties of operations

  • Commutative

  • Associative

  • Distributive

In order to meet these essential understandings, students must be able to...

• Add and subtract multi-digit whole numbers fluently and precisely with the standard algorithm

• Multiply up to a 4-digit number by a one-digit number

• Multiply two 2-digit numbers

• Divide a 4-digit whole number by a 1-digit whole number

• Explain multiplication calculations using equations, rectangular arrays, and/or area models

• Use strategies/properties of operations to solve multi-digit arithmetic problems

• To engage in respectful mathematical discourse when exploring math concepts

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Explain why a fraction a/b is equivalent to a nxa /nxb 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. (CCSS: 4.NF.A.1)

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. 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, e.g., by using a visual fraction model. (CCSS: 4.NF.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Use the equivalence and/or comparative value of fractions to solve problems, applying a variety of strategies

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• The role of the numerator, denominator, part, and whole

• Different fractions can have the same value (equivalency)

In order to meet these essential understandings, students must know...

Academic Vocabulary: fraction, numerator, denominator, unit fraction, equivalent

• When a numerator and denominator of a fraction are multiplied by the same number, the resulting fraction is equivalent to the original fraction

• Comparisons of fractions are valid only when the two fractions refer to the same whole

• How to use multiple strategies and tools to compare fractions and develop equivalency

In order to meet these essential understandings, students must be able to...

• Recognize and generate equivalent fractions

• Discuss fraction equivalency and comparisons using the terms: numerator, denominator, part, and whole

• Explain why a fraction is equivalent to another

• Use visual models to compare fractions

• Articulate and justify conclusions when comparing fractions

• Order and compare fractions

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (CCSS: 4.NF.B.3)

   • Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. (CCSS: 4.NF.B.3.a)

   • Decompose a fraction into a sum of fractions with like denominators in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8, 3/8 = 1/8 + 2/8, 2 1/8 = 1 + 1 + 1/8= 8/8 + 8/8 + 1/8. (CCSS: 4.NF.B.3.b)

   • Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. (CCSS: 4.NF.B.3.c)

   • Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. (CCSS: 4.NF.B.3.d)

2. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (CCSS: 4.NF.B.4)

   • Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x 1/4, recording the conclusion by the equation 5/4 = 5 x 1/4. (CCSS: 4.NF.B.4.a)

   • Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x 2/5 as 6 x 1/5}, recognizing this product as 6/5. (In general, n x a/b = nxa/b.) (CCSS: 4.NF.B.4.b)

   • Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? (CCSS: 4.NF.B.4.c)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Use an understanding of the structure of fractions to perform operations with fractions

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Addition and multiplication of fractions as joining parts referring to the same whole

• Subtraction of fractions as separating parts referring to the same whole

• Recognize the mathematical connections between the indicated operations with fractions and the corresponding operations with whole numbers

• Explain how the operations connect to the structure of fractions

In order to meet these essential understandings, students must know...

Academic Vocabulary: fraction, numerator, denominator, mixed number, sum, difference

• Fraction a/b is a multiple of 1/b

• Relationship between written fractions and visual models of fractions

• Relationship between visual fraction models and equations

• How multiplying two whole numbers relates to multiplying a fraction by a whole number

In order to meet these essential understandings, students must be able to...

• Multiply a fraction by a whole number

• Add and subtract fractions and mixed numbers with like denominators

• Decompose a fraction into a sum of fractions with like denominators

• Solve word problems involving addition and subtraction of fractions with like denominators and multiplying a fraction by a whole number

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. 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. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. (CCSS: 4.NF.C.5)

2. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. (CCSS: 4.NF.C.6)

3. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. (CCSS: 4.NF.C.7)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Fluently use comparisons to solve problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• The relationship between place value and fractions with a denominator for 10 or 100

• Convert between fraction and decimal form (denominator of 10 and 100) to perform comparisons and solve problems

• Drawings, models, and manipulatives can be used to represent decimals and fractions

In order to meet these essential understandings, students must know...

Academic Vocabulary: tenths, hundredths, decimal, convert

• The meaning of tenths and hundredths place values

• Both fractions and decimals represent parts of a whole

• 0.7 = 7/10 = 70/100 (example)

• 0.38 = 38/100 (example)

• Fractions with like denominators can be compared, added, and subtracted

• Decimal notation for fractions

In order to meet these essential understandings, students must be able to...

• Convert between fractions and decimals as described in the “must know” section

• Compare two decimals to hundredths by reasoning about their size

• Express a fraction with denominator 10 as an equivalent fraction with a denominator 100

• Add fractions with denominators 10 and 100

• Use place value structure to compare and express decimal numbers to the tenths and hundredths

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. (CCSS: 4.OA.A.1)

2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (See Appendix, Table 2) (CCSS: 4.OA.A.2)

3. 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. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (CCSS: 4.OA.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Use mathematics to model real-world problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• The roles of the four operations in real-world problems

• Use of symbols, such as letters, to represent unknown values to write and solve equations

In order to meet these essential understandings, students must know...

• Difference between multiplicative and additive comparisons

• Strategies for computing using the 4 operations

In order to meet these essential understandings, students must be able to...

• Represent verbal statements of multiplicative comparisons as multiplication equations

• Solve multi-step word problems using the four operations and whole numbers

• Use estimating and mental computation to assess the reasonableness of answers

• Contextually interpret remainders in division word problems

• Use drawings and equations to solve problems

• Make sense of multi-step word problems to understand the relationship between known and unknown quantities

• Reason quantitatively with word problems

• Make sense of problems and persevere in solving them

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. (CCSS: 4.OA.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Use relationships to solve problems and for mathematical reasoning.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• The regularities in determining whether a given number is a multiple of a given one-digit number

• The regularities in determining if a given number is prime or composite

• That a number is a multiple of each of its factors

• Relationships between factors and multiples can be used to solve problems and for mathematical reasoning

In order to meet these essential understandings, students must know...

Academic Vocabulary: factor, multiple, prime, composite

• Factor pairs for a whole number in the range 1-100

• Whole numbers are a multiple of each of its factors

• Difference between prime and composite numbers

In order to meet these essential understandings, students must be able to...

• Determine if a number 1-100 is prime or composite

• Determine whether a given whole number in the range of 1-100 is a multiple of a given one-digit number

• Describe reasoning when using multiples or factors when solving a problem

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. (CCSS: 4.OA.C.5)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Look for and express regularity in repeated reasoning

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Sequences of numbers or shapes can be described mathematically

In order to meet these essential understandings, students must know...

Academic Vocabulary: pattern, sequence, sequential

• That sequences of numbers or shapes can be governed by rules

• Some features of a pattern may not be explicit in the rule itself

• Use visuals to represent a mathematical rule

In order to meet these essential understandings, students must be able to...

• Generate a number or shape pattern that follows a given rule

• Notice when calculations are repeated and describe patterns in generalized, mathematical ways.

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb., oz.; l, ml; hr., min, sec. 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. For example, know that 1 ft. is 12 times as long as 1 in. Express the length of a 4 ft. snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,12), (2,24), (3,36), … (CCSS: 4.MD.A.1)

2. 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. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. (CCSS: 4.MD.A.2)

3. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. (CCSS: 4.MD.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Convert measurements to solve real-world problems

• Use appropriate tools to deepen understanding of mathematical concepts

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Solve problems involving measurement and unit conversion

• Make sense of quantities, their units, and their relationships in problem solving situations

In order to meet these essential understandings, students must know...

Academic Vocabulary: area, perimeter, convert

• Relative sizes of measurements units within one measurement system

• Area and perimeter formulas

• How to generate and use conversion tables to aid in measurement conversions and represent measurement quantities

• Convert between larger and smaller units of single measurement system

In order to meet these essential understandings, students must be able to...

• Convert between larger and smaller units within a single system of measurement

• Record measurement equivalents in a two-column table

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

• Including converting larger units to smaller units

• Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale

• Solve real-world area and perimeter problems

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Make a line plot to display a data set of measurements in fractions of a unit (1/2, ¼, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. (CCSS: 4.MD.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Represent and interpret data

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• The meaning of line plot data

• Reading and representing measurements recorded on line plots

In order to meet these essential understandings, students must know...

• Relative value of different fractions

• Structure of a line plot

• Why it's helpful to organize data in line plots

• Why it’s important to establish the whole when plotting fractions on a number line

• How labels help the reader determine importance and understand data

In order to meet these essential understandings, students must be able to...

• Make a line plot to display a data set of measurements in fractions of a unit

• Solve problems involving addition and subtraction of fractions by using information presented in line plots

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: (CCSS: 4.MD.C.5)

   • 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 “one-degree angle,” and can be used to measure angles. (CCSS: 4.MD.C.5.a)

   • An angle that turns through n one-degree angles is said to have an angle measure of n degrees. (CCSS: 4.MD.C.5.b)

2. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. (CCSS: 4.MD.C.6)

3. 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, e.g., by using an equation with a symbol for the unknown angle measure. (CCSS: 4.MD.C.7)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Analyze and measure the size of angles in real-world and mathematical problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Reason abstractly and quantitatively about angles and angular measurement

In order to meet these essential understandings, students must know...

• Angles are geometric shapes formed by 2 rays that share a common end point

• Relationship between degrees and the part of a circle encompassed by an angle with its vertex at the center of the circle

• Angle measures are additive: when an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures as parts

In order to meet these essential understandings, students must be able to...

• Use a protractor to measure and sketch angles

• Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. (CCSS: 4.G.A.1)

2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. (CCSS: 4.G.A.2)

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. (CCSS: 4.G.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

Categorizing enables the comparison of properties to identify lines, shapes, and symmetry

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

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

In order to meet these essential understandings, students must know...

Academic Vocabulary: symmetry, point, line, line segment, ray, perpendicular line, parallel lines, right angle, obtuse, acute

• Lines of symmetry

• Define and recognize: point, line, line segment, ray, types of angles, perpendicular and parallel lines

• Make observations and draw conclusions about the classification of two-dimensional figures based on the absence or presence of specified attributes

In order to meet these essential understandings, students must be able to...

• Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines

  • Identify these in 2-dimensional figures

• Classify 2-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size

• Identify right triangles

• Draw lines of symmetry

• Recognize a line of symmetry

• Identify line-symmetric figures

• Justify and reason to support claims as a mathematician