Aligning NJ Grade 5 Mathematics Curricula to the Common Core State Standards
NEW / OLDCommon Core State Standards (CCSS) adopted June 16, 2010 / How is it related to the old content? / 2008 NJ Core Curriculum Content Standards (NJ cccs) / If not related, where did old content go?
Operations and Algebraic Thinking 5.OA
Write and interpret numerical expressions.
5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. / CCSS move this from grade 6 (NJcccs 4.1.6.B.8) to grade5. / NEW (to grade 5) / CCSS 6.EE.2 builds upon this work with expressions.
5.OA.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 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. / Somewhat similar but more specific and narrower expectation, in that expressions with variables are not included in the new CCSS / 4.3.5.C. 1. Use number sentences to model situations.
- Using variables to represent unknown quantities
- Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations
4.3.5.C.2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.
- Changes over time
- Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)
[This is related to content that will now be learned in grade 8.]
4.3.5.D.1. Solve simple linear equations with manipulatives and informally
- Whole-number coefficients only, answers also whole numbers
- Variables on one side of equation
[This is related to content that will now be learned in grade 6.]
Analyze patterns and relationships.
5.OA.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. 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. / Similar but slightly more specific expectation / 4.3.5.A.1. Recognize, describe, extend, and create patterns involving whole numbers.
- Descriptions using tables, verbal rules, simple equations, and graphs
4.3.5.B.1. Describe arithmetic operations as functions, including combining operations and reversing them. / “Functions” are not explicitly introduced in the new CCSS until Grade 8.
4.3.5.B.2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.
Number and Operations in Base Ten 5.NBT
Understand the place value system.
5.NBT.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 move this from grade 4 (NJcccs 4.1.4.A.2) to grade5. / NEW (to grade 5) / In Transition:
In many 5th grade classes, this will not need to be taught as new material the first year, as it would have been included in grade 4 understanding of place value (CCSS4.NBT.1)
5.NBT.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 10. / CCSS move this from grade 6 (NJcccs 4.1.6.B.2) to grade5. Note that this includes use of exponents. / NEW (to grade 5)
5.NBT.3. Read, write, and compare decimals to thousandths. / Similar but slightly more specific expectation / 4.1.5.A.1, 4.1.5.A.3, and 4.1.5.A.6 below
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). / Similar but slightly more restricted expectation (to thousandths) / 4.1.5.A.1. Use real-life experiences, physical materials, and technology to construct meanings for numbers.
- All decimals
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. / Similar. Note that the use of > and < symbols is not new, but rather grade1 (CCSS1.NBT.3). / 4.1.5.A.6. Compare and order numbers. [Including all fractions and decimals, from 4.1.5.A.1]
4.1.5.A.3. Demonstrate a sense of the relative magnitudes of numbers (as applied to fractions and decimals). / Not explicitly articulated in the CCCS at this grade
5.NBT.4. Use place value understanding to round decimals to any place. / Related but narrower, more specific expectation / 4.1.5 C.1. Use a variety of estimation strategies for both number and computation.
4.1.5.A.5. Develop and apply number theory concepts in problem solving situations.
- Primes, factors, multiples
In Transition: Students coming to fifth grade from classes in which the 2008 standards were used may have not yet mastered the content of CCSS4.OA.4.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
4.1.5.B.1. Recognize the appropriate use of each arithmetic operation in problem situations. / Not explicitly included in CCSS at this grade.
5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm. / This is a new expectation that builds on students multiplying 2-digit numbers in grade4. / NEW (both because
it goes beyond two-digit numbers and because it specifies a particular algorithm) / For classes in which the standard pencilandpaper algorithm is being applied to multiplying three- and fourdigit numbers, this would not be new.
5.NBT.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. / Similar / 4.1.5.B.3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.
4.1.5 B.6. Understand and use the various relationships among operations and properties of operations. / In the new CCSS, this receives greater emphasis in grade 6
5.NBT.7. 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. / Similar for adding and subtracting decimals / 4.1.5.B.2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:
- Pencil-and-paper
- Mental math
- Calculator
In Transition: Students coming to fifth grade from classes in which the 2008 standards were used may have not yet mastered the content of CCSS4.NF.3.
Multiplying and dividing are new expectations for decimals at grade 5. In the 2008 NJ cccs, multiplication and division of decimals were in grade 6. / NEW (to grade 5)
Money is a familiar concrete model for decimals to hundredths, although not explicitly articulated in the new CCSS. / 4.1.5.A.2. Recognize the decimal nature of United States currency and compute with money. / Although included in the CCSS explicitly only at grade 4, computation with money is an important application of computation throughout the grades.
4.1.5.B.4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers. / In the new CCSS, selection of a method is not explicitly included at this grade. It receives greater emphasis in the grade 4 introduction.
Number and Operations—Fractions 5.NF
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.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 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) / Similar but slightly more specific expectation. Note particularly that the new CCSS explicitly includes both unlike denominators and mixed numbers. / 4.1.5.A.4. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.
4.1.5.B.2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:
- Pencil-and-paper
- Mental math
- Calculator
5.NF.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. / Similar / 4.1.5.B.1. Recognize the appropriate use of each arithmetic operation in problem situations.
4.5.5.A.1. Learn mathematics through problem solving, inquiry, and discovery.
4.5.5.E.1. Create and use representations to organize, record, and communicate mathematical ideas.
- Concrete representations (e.g., base-ten blocks or algebra tiles)
- Pictorial representations (e.g., diagrams, charts, or tables)
- Symbolic representations (e.g., a formula)
- Graphical representations (e.g., a line graph)
4.1.5.B.5. Check the reasonableness of results of computations.
4.1.5.C.3. Determine the reasonableness of an answer by estimating the result of operations.
4.1.5.C.2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer. / Not explicitly articulated in the new CCSS at any grade
4.1.5.C.4. Determine whether a givenestimate is an overestimate or an underestimate.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.3. 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? / Similar / 4.1.5.A.1. Use real-life experiences, physical materials, and technology to construct meanings for numbers.
- All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers
4.1.5.A.6. Compare and order numbers.
5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. / The new CCSS move multiplication from grade 6 to grade 5. In the 2008 NJcccs, both multiplication and division of fractions were in grade 6. / NEW (to grade 5)
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. 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.
5.NF.5. Interpret multiplication as scaling (resizing), by: / In the 2008 CCCS, scaling was first mentioned in grade 6 / NEW (to grade 5)
a. 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 move this from grade 7 to grade 5. / NEW (to grade 5)
b. 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 = (n×a)/(n×b) to the effect of multiplying a/b by 1. / CCSS move this from grade 6 to grade 5. / NEW (to grade 5)
5.NF.6. 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 move this from grade 6 to grade 5. / NEW (to grade 5)
5.NF.7. 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.” (Footnote to Common Core State Standards)] / CCSS move this from grade 6 to grade 5. / NEW (to grade 5)
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story contextfor (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explainthat(1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. / CCSS move this from grade 6 to grade 5. / NEW (to grade 5)
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for4 ÷ (1/5), and use a visual fraction model to show the quotient. Usethe relationship between multiplication and division to explain that4 ÷ (1/5) = 20 because 20 × (1/5) = 4. / CCSS move this from grade 6 to grade 5. / NEW (to grade 5)
c. 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 eachperson get if 3 people share 1/2 lb of chocolate equally? How many1/3-cup servings are in 2 cups of raisins? / CCSS move this from grade 6 to grade 5. / NEW (to grade 5)
Measurement and Data 5.MD
Convert like measurement units within a given measurement system.
5.MD.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. / Similar / 4.2.5.D.2. Convert measurement units within a system (e.g., 3 feet = __ inches).
4.2.5.D.4. Use measurements and estimates to describe and compare phenomena. / Implied by the new CCSS solving of real-world problems
4.5.5.A.2. Solve problems that arise in mathematics and in other contexts.
- Open-ended problems
- Non-routine problems
- Problems with multiple solutions
- Problems that can be solved in several ways
4.2.5.D.3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile). / Not explicitly included in CCSS at any grade.
Represent and interpret data.
5.MD.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. For example,given different measurements of liquid in identical beakers, find theamount of liquid each beaker would contain if the total amount in all thebeakers were redistributed equally. / This is a new expectation that combines line plots from previous grades with operations on fractions from this grade. / NEW
4.4.5.A. Data Analysis / With the exception of the very narrow expectations of 5.MD.2, this Strand is not included in the new CCSS at this grade.
4.4.5.B. Probability / The Probability Strand is not included in the new CCSS at this grade.
4.4.5.C. Discrete Mathematics-
Systematic Listing and Counting and
/ The Discrete Mathematics Strands are not included in the new CCSS at this grade.4.4.5.D. Discrete Mathematics-Vertex-Edge Graphs and Algorithms
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.4.2.5.E.1. Use a protractor to measure angles. / Measuring angles is grade 4 content in the new CCSS (4.MD.5 through 7).
In Transition: Students coming to fifth grade from classes in which the 2008 standards were used may have not yet mastered the content of CCSS 4.MD.5, 6, & 7; and this may need to be temporarily taught in grade 5.
4.2.5.D.1. Select and use appropriate units to measure angles and area.
4.2.5.E.2. Develop and apply strategies and formulas for finding perimeter and area.
- Square
- Rectangle
[In the new CCSS, area and perimeter are learned in grades 3 and 4; volume is learned in grade 5]
4.2.5.E.3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.
4.2.5.E.4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area ofthe bottom of one’s foot).
5.MD.3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. / This is beyond the level of specificity provided in the 2008 NJcccs.
[In the new CCSS, area and perimeter are learned in grade 4; volume is learned in grade 5] / NEW
a. 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.
b. 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.
5.MD.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. / CCSS move this from grade 4 (NJ cccs 4.2.4.E.3) to grade5. / NEW (to grade 5)
5.MD.5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. / New expectation at grade 5. / NEW
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. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. / CCSS move this from grade 4 to grade5. This instructional guidance is beyond the level of specificity provided in the NJ cccs. It is related to 2008 expectations from 4.2.3.E.3 and 4.2.4.E.3 . / NEW (to grade 5) / In Transition: Students coming to fifth grade from classes in which the 2008 standards were used may have already experienced these activities and know this content. Once the curriculum change has been implemented at grade 3 and 4 in a district, teachers can no longer assume such previous familiarity.
b. Apply the formulas V = l × w × h and V = B × 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 move this from grade 6 (NJ cccs 4.2.6.E.3) to grade5. / NEW (to grade 5)
c. 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. / NEW
Geometry 5.G
Graph points on the coordinate plane to solve realworld and mathematical problems.
5.G.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 (e.g., x-axis and x-coordinate, y-axis and y-coordinate). / The new CCSS move introduction of the coordinate system (quadrant I) from grade 3 to grade 5. [In the future, students entering grade 5 should not be expected to have prior knowledge of the coordinate system.] / NEW (to grade 5) / In Transition: For the immediate future, students may only need to review what they learned previously. However, eventually, students entering grade 5 should not be expected to have prior knowledge of the coordinate system.
5.G.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. / Related (in terms of using the first quadrant on a coordinate grid), but with different emphases. / 4.2.5.C.1. Create geometric shapes with specified properties in the first quadrant on a coordinate grid.
Classify two-dimensional figures into categories based on their properties.
4.2.5 A.1. Understand and apply concepts involving lines and angles.
- Notation for line, ray, angle, line segment
- Properties of parallel, perpendicular, and intersecting lines
- Sum of the measures of the interior angles of a triangle is 180°
Much of bullet 2 is in grade 4 (4.G.1 & 2).
Sum of interior angles is grade 8 in the CCSS.
5.G.3. Understand 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 arerectangles, so all squares have four right angles. / The new CCSS move inclusive relationships (NJcccs 4.2.4.A.2) from grade 4 to grade 5. / NEW (to grade 5)
5.G.4. Classify two-dimensional figures in a hierarchy based on properties. / Similar, but much less specific than the 2008 expectation. / 4.2.5.A.2. Identify, describe, compare, and classify polygons.
- Triangles by angles and sides
- Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi
- Polygons by number of sides.
- Equilateral, equiangular, regular
- All points equidistant from a given point form a circle
4.2.5.A.3. Identify similar figures. / Postponed until grade 8 in CCSS.
4.2.5 A.4. Understand and apply the concepts of congruence and symmetry (line and rotational). / In the new CCSS, line symmetry is included at grade 4 (4.G.3). Congruence is postponed until grade 8. Rotational symmetry is not explicitly included in CCSS at any grade.
4.2.5.B. Transforming Shapes / These Strands are not included in the new CCSS at this grade.
4.4.5.C. Discrete Mathematics-
Systematic Listing and Counting
4.4.5.D.1. Devise strategies for winning simple games (e.g., start with two piles of objects, each oftwo players in turn removes any number of objects from a single pile, and the person totake the last group of objectswins) and express those strategies as sets of directions. / Not included in the new CCSS at any grade.Shaded content is not a focus in this grade because it has already been covered in previous grades, because it will be covered in later grades, or because it is not included at any grade in the new CCSS.