Evidence Outcomes:

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. (CCSS: 5.NBT.A.1)

2. 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. Use whole- number exponents to denote powers of

3. 10. (CCSS: 5.NBT.A.2)

4. Read, write, and compare decimals to thousandths. (CCSS: 5.NBT.A.3) Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 × 10 + 7 x 1 + 3 x 1/10 + 9 x 1/100 + 2 x 1/1000. (CCSS: 5.NBT.A.3.a) Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (CCSS: 5.NBT.A.3.b)

5. Use place value understanding to round decimals to any place. (CCSS: 5.NBT.A.4)

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

• Understand patterns within the place value system

• Use place value understanding to round numbers to any place in new contexts.

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

• The relationship between numbers such as 25, 2.5 and 0.25

• How fractions represent decimal place value

• Place value reasoning with whole numbers to decimals

• Place value is not just making tens with greater place value but making tenths with lesser place values

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

Academic Vocabulary: decimal, tenths, hundredths, thousandths, digit, value,

• Decimal place value to the thousandths

• A digit in one place value is 10 times greater than what it represents in the place to its right

• A digit in a place value is 1/10 of what it represents in the place to its left

• How to read and write numbers to the thousandths place using standard form, word form, unit form, and expanded form

• powers of 10

• Use whole-number exponents to denote powers of ten

• Use place value to round numbers to any place (decimals and whole numbers)

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

• Show visually and with an equation the relationship between numbers like 25, 2.5 and 0.25

• Compare two decimals to thousandths (>, <, =)

• Round decimals to any place

• read and write numbers to the thousandths place using standard form, word form, unit form , and expanded form

• Explain and connect patterns in the number of zeros in a product when multiplying powers of 10

• Justify and use reasoning to support mathematical claims

• Describe placement of a decimal point when a decimal is multiplied or divided by a power of 10

Evidence Outcomes:

1. Fluently multiply multi-digit whole numbers using the standard algorithm. (CCSS: 5.NBT.B.5)

2. 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.

3. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (CCSS: 5.NBT.B.6)

4. Add, subtract, multiply, and divide decimals to hundredths, 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 method and explain the reasoning used. (CCSS: 5.NBT.B.7)

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 processes students must understand are...

• Operations of multi-digit whole number and decimals to hundredths to solve problems

• Calculations are based on properties of operations, equations, drawings/arrays, or other models.

• Place value structure helps to compute with whole numbers and decimals

• Relationships between and within operations

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

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

• How to estimate to check for reasonableness

• How to multiply multi-digit whole numbers with the standard algorithm

• Use strategies to divide whole numbers with up to 4 digit dividends and two-digit divisors

• Use strategies to add and subtract decimals to the hundredths

• Use strategies to multiply and divide decimals to the hundredths

• Relationship between operations

• Properties of operations

  • Commutative

  • Associative

  • Distributive

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

• Fluently multiply multi-digit whole numbers with the standard algorithm

• Divide whole numbers with up to 4 digit dividends and two-digit divisors using a strategy to accurately and efficiently solve problems

• Add, subtract, multiply and divide decimals to hundredths using strategies to accurately and efficiently solve problems

• Estimate and assess the reasonableness of answers

• Justify and using reading to support mathematical claims

Evidence Outcomes:

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. For example, 2/3 + \5/4} = 8/12 + 5/12 = 23/12. (In general, a/b + c/d = ad +bc/bd.) (CCSS: 5.NF.A.1)

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. (CCSS: 5.NF.A.2)

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

Examine and apply the use of equivalent fractions to accurately and effectively solve addition and subtraction of fractions

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

• The reasonableness of fraction calculations by estimating results using benchmark fractions and number sense.

• The structure in the multiplicative relationship between unlike denominators when creating equivalent fractions.

• Equivalent fractions are a strategy to add and subtract fractions

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

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

• What equivalent fractions are

• Steps to create equivalent fractions

• Add and subtract fractions with unlike denominators using equivalent fractions

• Use benchmark fractions to estimate sums and differences of fraction problems

• Use visual fraction models or equations to represent a problem

• How to estimate with fractions

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

• Create equivalent fractions

• Add and subtract fractions with unlike denominators

• Solve word problems involving addition and subtraction of fractions including unlike denominators

• Estimate and assess th reasonableness of answers

• Justify and using reading to support mathematical claims

Evidence Outcomes:

1. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4} multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? (CCSS: 5.NF.B.3)

2. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. (CCSS: 5.NF.B.4) Interpret the product a/b x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. For example, use a visual fraction model to show 2/3 x 4 = 8/3, and create a story context for this equation. Do the same with 2/3 x 4/5= 8/15. (In general, a/b x c/d = ac/bd.) (CCSS: 5.NF.B.4.a) Find the area of a rectangle with fractional side lengths by tiling it with unit squares 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. (CCSS: 5.NF.B.4.b)

3. Interpret multiplication as scaling (resizing), by: (CCSS: 5.NF.B.5) Comparing 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. (CCSS: 5.NF.B.5.a) Explaining 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); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = nxa/nxb to the effect of multiplying a/b by 1. (CCSS: 5.NF.B.5.b)

4. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (CCSS: 5.NF.B.6)

5. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.) (CCSS: 5.NF.B.7) Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for 1/3 ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 1/3 ÷ 4 = 1/12 because 1/12 x 4 = 1/3. (CCSS: 5.NF.B.7.a) Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ 1/5, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ 1/5 = 20 because 20 x 1/5= 4. (CCSS: 5.NF.B.7.b) Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? (CCSS: 5.NF.B.7.c)

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

• Apply and extend previous understandings of multiplication and division

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

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

• How to solve problems requiring calculations that scale whole numbers and fractions

• Fraction models and arrays to interpret and explain fraction calculations

• The underlying unit quantities when solving problems involving multiplication and division of fractions

• Multiplying fractions by a whole number greater than 1 can result in a product less than the whole number

• Interpret a fraction as division of the numerator by the denominator

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

Academic Vocabulary: fraction, numerator, denominator, unit fraction, equivalent, mixed number, product, scaling

• How to rewrite a fraction such as 5/ 3 as an addition equation

• How to rewrite a fraction 5/3 as a multiplication equation

• Understand that (𝑎/𝑏 = 𝑎 ÷ b)

• How to solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers

• Find the area of a rectangle with fractional side lengths by tiling it with unit squares 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.

• Interpret multiplication as scaling

• Multiply fractions and mixed numbers

• Divide unit fractions by whole numbers

• Divide whole numbers by unit fractions

• How to use visual fraction models and equations to represent and solve problems

• How to estimate to check for reasonableness

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

• Interpret a fraction as division of the numerator by the denominator

• multiply a fraction by a fraction of a whole number by a fraction

• Explaining 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); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number

• Solve real-world problems involving multiplication of fractions and mixed numbers, and division of unit fractions by whole numbers, and division of the whole number by unit fractions

• Justify and using reading to support mathematical claims

Evidence Outcomes:

1. Use grouping symbols (parentheses, brackets, or braces) in numerical expressions, and evaluate expressions with these symbols. (CCSS: 5.OA.A.1)

2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 x (8 + 70. Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (CCSS: 5.OA.A.2)

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

• Write and interpret numerical expressions

• Based on an understanding of any problem, initiate a plan, execute it, and evaluate the reasonableness of the solution

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

• Expressions represent mathematical relationships between quantities

• Apply an understanding of structures that make the order of operations clear when reading and writing mathematical expressions

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

Academic Vocabulary: variable, expressions

• The communicative property and which operations it applies and does not

• The associative property and which operation it applies and does not

• The distributive property and which operation it applies and does not

• The “order of operations” math uses and why it’s needed

• What PEMDAS means

• letters represent unknown variables (These letters are actually numbers in disguise)

• Ways to write multiplication

  • a x b

  • a ᐧ b

  • a*b

  • (a) (b)

  • a (b)

• (a) b

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

• Use grouping symbols(parentheses, brackets, or braces)/ PEMDAS in numerical expressions

• Evaluate expressions using grouping symbols/PEMDAS

• Write simple expressions

• Interpret numerical expressions without evaluating

• Justify and using reading to support mathematical claims

Evidence Outcomes:

1. 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. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. (CCSS: 5.OA.B.3)

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

• Analyze patterns and relationships

• 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...

• Analyzing and comparing patterns

• Reasoning quantitatively with patterns by relating sequences of numbers with the rule that generated them

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

• How to generate patterns using given rules

• Corresponding terms

• Ordered pairs

• How to form ordered pairs

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

• Graph corresponding terms formed by two numerical rules

• Generate two numerical patterns using two given rules

• Form ordered pairs consisting of corresponding terms from two patterns

• Graph ordered pairs on a coordinate plane

• Explain patterns and mathematical relationships in/with patterns

• Justify and using reading to support mathematical claims

Evidence Outcomes:

1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems. (CCSS: 5.MD.A.1)

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...

• Convert like measurements units within a given measurement system

• Use appropriate precision when converting measurements based on a problem's context

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

• How to convert from larger to smaller units within a system(US Customary and Metric)

• How to convert from smaller units to larger units within a system(US Customary and Metric)

• How to convert from larger to smaller units across systems(US Customary and Metric)

• How to convert from smaller units to larger units across systems(US Customary and Metric)

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

• Convert larger units to small units within and across systems(US Customary and Metric)

• Convert small units to larger units within and across systems(US Customary and Metric)

• Use conversions to solve multi-step real-world problems

• Justify and using reading to support mathematical claims

Evidence Outcomes:

1. 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. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. (CCSS: 5.MD.B.2)

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

• Represent and interpret data

• Analyze data to seek out patterns and/or make predictions

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

• Line plots with fractional measurement data

• Measurement data presented in line plots to participate in a discussion about the data

• Scaling of line plots to represent fractional measurements

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

• Structure of a line plot

• How to create line plots

• Fractional measurements

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

• Establishing 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 data sets of measurements in fractions of a unit

• Solve problems involving operations of fractions by using information presented in line plots

• Justify and reason to support claims as a mathematician

Evidence Outcomes:

1. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. (CCSS: 5.MD.C.3)A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume and can be used to measure volume. (CCSS: 5.MD.C.3.a) 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. (CCSS: 5.MD.C.3.b)

2. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft., and improvised units. (CCSS: 5.MD.C.4)

3. Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume. (CCSS: 5.MD.C.5) Model 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. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. (CCSS: 5.MD.C.5.a) Apply the formulas V = I x w x h and V = b x 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. (CCSS: 5.MD.C.5.b) Use the additive nature of volume to 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. (CCSS: 5.MD.C.5.c)

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

• Understand concepts of volume and relate volume to multiplication and addition

• Examine and apply a variety of strategies to accurately and efficiently solve problems

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

• Real-world problems involving volume

• Connections between the values being multiplied in a volume formula, the concept of cubic units, and the context within which volume is being calculated

• The structure of two-dimensional space and the relationship between arrays and area to three-dimensional space and the relationship between layers of cubes and volume

• Concepts of volume and relate volume to multiplication and addition

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

Academic vocabulary: volume, rectangular prism

• Volume is an attribute of solid figures

• How volume and area are related

• A cube with a side length of 1 unit is a “unit cube” and can be used to measure the volume

• Associative property of multiplication

• Volume as an operation of multiplication and addition

• Model volume of right rectangular prisms by packing it with unit cubes and make connections to multiplying edge lengths equivalently (V=l x w x h)or multiplying the height by the area of the base (V=b x h)

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

• Measure volumes using unit cubes

  • Cubic cm, cubic in, cubic ft, and improvised units(water in a fish tank, sand in a box)

• Apply the formulas V=l x w x h and V=b x h for rectangular prisms with whole number edge lengths

• Solve real-world problems involving volume

Evidence Outcomes:

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.

2. 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 (e.g., x-axis and x-coordinate, y-axis and y-coordinate). (CCSS: 5.G.A.1)

3. 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. (CCSS: 5.G.A.2)

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

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

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

• Information presented visually in a coordinate plane

• The first quadrant of the coordinate plane is a tool to represent, analyze, and solve problems

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

Academic vocabulary: coordinate plane, axes, ordered pairs, quadrant

• Define a coordinate plane(system)

• Definition of axes(x, y)

• Ordered pairs

• Plot ordered pairs

• Identify a location on a coordinate plane(first quadrant) using ordered pairs

• Understand that the first number indicates how far to travel from the origin in the direction of one axis(x), and the second number indicates how far to travel in the direction of the second axis(y)

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

• 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.

• Explain what things in the real world are designed like a coordinate plane or that use a coordinate system

Evidence Outcomes:

1. Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. (CCSS: 5.G.B.3)

2. Classify two-dimensional figures in a hierarchy based on properties. (CCSS: 5.G.B.4)

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

• Categorizing enables the comparison of properties to identify two-dimensional figures

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

• Attributes of two-dimensional figures to classify them

• Measurement tools help improve the classification of shapes

• Attributes of two-dimensional shapes to classify the shapes in a hierarchy of figures

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

• Definition of two-dimensional figures

• Properties and attributes of two-dimensional figures

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

• Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category

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

• Use the words “always,” “sometimes,” and “never” to develop a classification of two-dimensional figures