Aligning NJ Grade 8 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?
The Number System 8.NS
Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.1. Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational. / CCSS move the fuller consideration of irrational numbers from HS to grade 8. / 4.1.8.A.7. Construct meanings for common irrational numbers, such as (pi) and the square root of 2. / Percents get more attention in grade 7 in the CCSS.
4.1.8.A.1. Extend understanding of the number system by constructing meanings for the following:
- Rational numbers
- Percents
4.1.8.A.6. Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.
- 5/7 = 0. 714285714285… = 0.
8.NS.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. / CCSS move the fuller inclusion of irrational numbers from HS to grade8. / 4.1.8.A.4. Compare and order numbers of all named types
- Rational numbers
- Percents
- Exponents
- Roots
- Absolute values
- Numbers represented in scientific notation
Scientific notation is in grade 8 in CCSS 8.EE.4 below.
4.1.8.C.1. Estimate square and cube roots of numbers.
[“Construct meanings for common irrational numbers” from 4.1.8.A.7 above]
4.1.8.C.3. Recognize the limitations of estimation and assess the amount of error resulting from estimation. / Although not explicitly articulated in CCSS at any grade this is implicit in CCSS 8.NS.2.
4.1.8.A.3. Understand and use ratios, rates, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations. / Deserving of more attention in grade 7 in the CCSS.
In Transition: Students coming to eighth grade from classes in which the 2008 standards were used may not have developed sufficient familiarity with ratios, rates,and proportions; this may therefore temporarily still need some attentionin grade 8.
4.1.8.A.5. Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number. [A duplicate of 4.1.7.A.5] / Deserving of more attention in grade 7 in the CCSS.
4.1.8.C.2. Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. [A duplicate of 4.1.7.C.1] / Deserving of more attention in grade 7 in the CCSS.
Expressions and Equations 8.EE
Work with radicals and integer exponents.
8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5= 3–3= 1/33 = 1/27. / CCSS move the inclusion of negative exponents from HS to grade 8. / 4.1.8.B.2. Use exponentiation to find whole number powers of numbers.
4.1.8.B.1. Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:
- Pencil-and-paper
- Mental math
- Calculator
8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x2= p and x3= p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. / Similar / 4.1.8.A.1. Extend understanding of the number system by constructing meanings for the following:
- Exponents
- Roots
4.1.8.B.3. Find square and cube roots of numbers and understand the inverse nature of powers and roots.
4.1.8.A.7. Construct meanings for common irrational numbers, such as (pi) and the square root of 2.
8.EE.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. / Related but much more specific expectation / 4.1.8.A.2. Demonstrate a sense of the relative magnitudes of numbers (as applied to exponents and numbers represented in scientific notation).
8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. / Similar but more specific expectation / 4.1.8.A.1. Extend understanding of the number system by constructing meanings for the following:
- Numbers represented in scientific notation
4.2.8.D.4. Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.
[Related to Mathematical Process #6 description, that students “calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.”] / Although not separately articulated in a CCSS content standard, this is an important part of the process for students at this grade.
4.5.8.F.6. Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).
4.3.8 D.1. Use graphing techniques on a number line.
- Absolute value
- Arithmetic operations represented by vectors (arrows)
In CCSS, the focus on operations with integers is in grade 7; vectors are in HS.
4.3.8 D.4. Create, evaluate, and simplify algebraic expressions involving variables.
- Order of operations, including appropriate use of parentheses
- Distributive property
- Substitution of a number for a variable
- Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa
In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not have developed sufficient familiarity with the distributive property; this may therefore need some attention.
4.1.8 B.4. Solve problems involving proportions and percents. / In CCSS, this is grade7 content (7.RP.3).
4.1.8 B.5. Understand and apply the standard algebraic order of operations, including appropriate use of parentheses. / In CCSS, this is grade6 content (6.EE.2c) (except for exponents, which is grade7 content).
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. / Somewhat similar, but CCSS is narrower and more specific than the NJcccs. This one covers linear functions where the vertical intercept is 0. / 4.3.8.B.1. Graph functions, and understand and describe their general behavior.
- Equations involving two variables
- Rates of change (informal notion of slope)
8.EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. / This CCSS is very specific and not explicitly articulated in the NJcccs. It covers linear functions where the vertical intercept is not equal to 0.
4.3.8.B.2. Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations. / In CCSS, differentiating between linear and exponential growth is postponed until HS.
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). / Similar, but more specific expectation. / 4.3.8.D.2. Solve simple linear equations informally, graphically, and using formal algebraic methods.
- Multi-step, integer coefficients only (although answers may not be integers)
- Simple literal equations (e.g., A = lw)
- Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. / Somewhat similar, but the CCSS include equations with fractional coefficients / In CCSS, the various methods (including technology) are not explicitly delineated prior to HS.
4.3.8.D.3. Solve simple linear inequalities. / In CCSS, there is a focus on inequalities in grade7 and in HS
4.3.8.D.5. Understand and apply the properties of operations, numbers, equations, and inequalities.
- Additive inverse
- Multiplicative inverse
- Addition and multiplication properties of equality
- Addition and multiplication properties of inequalities
multiplicative inverse is a focus in HS;
properties of equality are used throughout to solve equations;
there is a focus on inequalities in grade7 and in HS.
8.EE.8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. / This CCSS moves solving systems of equations from HS to grade8 / NEW (to grade 8)
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x+2y cannot simultaneously be 5 and 6. / This CCSS moves solving systems of equations from HS to grade8 / NEW (to grade 8)
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. / This CCSS moves solving systems of equations from HS to grade8 / NEW (to grade 8)
4.3.8 A.1. Recognize, describe, extend, and create patterns involving whole numbers, rationalnumbers, and integers. / Without the formal terminology, sequences are introduced in grades 4 and 5 (5.OA.3) in the CCSS. The formal study of arithmetic and geometric sequences is postponed until HS.
- Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions
- Finite and infinite sequences
- Arithmetic sequences
- Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)
- Generating sequences by using calculators to repeatedly apply a formula
Functions 8.F
Define, evaluate, and compare functions.
8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) / The new CCSS introduce the concept of a function in grade 8, rather than assuming previous exposure to the concept. The new standards include the expectations of the 2008 NJ cccs for grade 8. / 4.3.8.B.1. Graph functions, and understand and describe their general behavior.
- Equations involving two variables
- Rates of change (informal notion of slope)
8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. / Similar / 4.3.8.C.2. Use patterns, relations, symbolic algebra, and linear functions to model situations.
- Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/ equations/inequalities
- Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standards and social studies standards)
8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. / Related, but different in emphases. The articulated focus in the CCSS is more precisely found in grade 6 NJcccs 4.3.6.B.1. / 4.3.8.C.1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations. / In Transition: Students coming to eighth grade from classes in which the 2008 standards were used may already know this content. Once the curriculum change has been implemented at grade 6 in a district, teachers can no longer assume previous familiarity with functions.
Geometry 8.G
Understand congruence and similarity using physical models, transparencies, or geometry software.
4.2.8.A.4. Understand and apply the concept of similarity.
- Using proportions to find missing measures
- Scale drawings
- Models of 3D objects
In Transition: Students coming to seventh grade from classes in which the 2008 standards were used may not have developed sufficient familiarity with scale drawings; they may therefore need some attention.
8.G.1. Verify experimentally the properties of rotations, reflections, and translations: / Similar to bullet 3 of the NJcccs, / 4.2.8.B.1. Understand and apply transformations.
- Finding the image, given the pre-image, and vice-versa
- Sequence of transformations needed to map one figure onto another
- Reflections, rotations, and translations result in images congruent to the pre-image
- Dilations (stretching/shrinking) result in images similar to the pre-image
a. Lines are taken to lines, and line segments to line segments of the same length. / Bullet 2 is included in CCSS 8.G.2 below
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines. / Dilations are included in CCSS 8.G.3 and 4 below.
8.G.2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. / Similar / [“Sequence of transformations needed to map one figure onto another” from 4.2.8.B.1.]
4.2.8.B.2. Use iterative procedures to generate geometric patterns.
- Fractals (e.g., the Koch Snowflake)
- Self-similarity
- Construction of initial stages
- Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle)
4.2.8.C.1. Use coordinates in four quadrants to represent geometric concepts. / This general statement overlaps 4.2.8.C.2 below and is duplicative of the grade 7 CPI.
8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. / Similar / 4.2.8.C.2. Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units). / In the 2008 NJ cccs, students were introduced to this in grade 7 (4.2.7.C.2). Once the new CCSS are implemented, students may arrive in grade 8 without such earlier exposure.
Related, but NJcccs was more specific / 4.2.8.E.1. Develop and apply strategies for finding perimeter and area.
- Impact of a dilation on the perimeter and area of a 2dimensional figure
8.G.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. / Similar, except that teachers may find that approaching congruence and similarity through transformations is a big change instructionally. / [“Sequence of transformations needed to map one figure onto another” and “Dilations (stretching/shrinking) result in images similar to the pre-image” from 4.2.8.B.1above]
8.G.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argumentin terms of transversals why this is so. / Very specific instructional guidance, beyond the level of specificity provided in the NJ cccs. CCSS move the sum of exterior angles from high school to grade8. / 4.2.8.A.3. Understand and apply properties of polygons.
- Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi
- Regular polygons
- Sum of measures of interior angles of a polygon
- Which polygons can be used alone to generate a tessellation and why
4.2.8.A.5. Use logic and reasoning to make and support conjectures about geometric objects. / Related to Mathematical Process No.3 description, that students “make conjectures and build a logical progression of statements to explore the truth of their conjectures.”