Wayne County Public Schools DRAFT August, 2012
Common Core State Standards
Curriculum Guide for Grade 8 Mathematics
Grade 8 Overview
· The Number System
o Know that there are numbers that are not rational, and approximate them by rational numbers --- understand that every number has a decimal expansion; use the decimal expansion to determine if a number is rational or irrational; use rational approximations to compare irrational numbers and to locate irrational numbers on a number line diagram.
· Expressions and Equations
o Work with radicals and integer exponents – integer exponents can be positive or negative; solve equations using square root and cube root symbols; use numbers expressed as a single digit times an integer power of 10 to estimate very large or very small quantities; perform operations with numbers expressed in scientific notation.
o Understand the connections between proportional relationships, lines, and linear equations -- graph proportional relationships, interpreting the unit rate as the slope of the graph; use similar triangles to understand slope between any 2 distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line going through the origin and the equation y=mx + b for a line intercepting the vertical axis at b.
o Analyze and solve linear equations and pairs of simultaneous linear equations – solve linear equations in 1 variable; analyze & solve
pairs of simultaneous linear equations ( systems of 2 linear equations in 2 variables).
· Functions
o Define, evaluate, and compare functions – determine input & output; compare 2 functions algebraically, graphically, numerically in tables,
or by verbal descriptions; identify linear & nonlinear functions.
o Use functions to model relationships between quantities – analyze the graphs to determine where the function is increasing or decreasing, if linear or nonlinear, etc.; sketch a graph that exhibits the qualitative features of a function that have been described verbally.
· Geometry
o Understand congruence and similarity using physical models, transparencies, or geometry software – verify congruence by using a sequence of rotations, reflections, and translations; describe similarity by using a sequence of rotations, reflections, translations, and dilations; 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.
o Understand and apply the Pythagorean Theorem –explain a proof and its converse; find the unknown side length in a right triangle; find the distance between 2 points in a coordinate system.
o Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres – know the formulas.
· Statistics and Probability
o Investigate patterns of association in bivariate data – construct and interpret scatter plots to investigate patterns of association between 2 quantities -- clustering, outliers, positive or negative association, linear or nonlinear association; interpret the slope and intercept of the equation of a linear model; investigate bivariate categorical data by displaying frequencies and relative frequencies in a 2-way table .
Resources:
NC DPI COMMON CORE INSTRUCTIONAL SUPPORT TOOLS Home Page: http://www.dpi.state.nc.us/acre/standards/common-core-tools/#unmath
NC DPI Grade 8 Math Unpacking Document: http://www.dpi.state.nc.us/docs/acre/standards/common-core-tools/unpacking/math/8th.pdf
NC DPI Grade 8 Math Curriculum Crosswalk: http://www.dpi.state.nc.us/docs/acre/standards/common-core-tools/crosswalks/math/grade8.pdf
Textbook Resource: Holt Middle School Math, Course 3, North Carolina Edition by Holt, Ó 2004
The Common Core State Standards Home Page: http://www.corestandards.org/
The Common Core State Standards for Mathematics: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
NC DPI Grade 8 Quick Reference Guide: http://aplus.ncdpi.wikispaces.net/file/view/ReferenceGuide_8th_grade.pdf
North Carolina
Timeline for Implementation
Common Core State Standards: Mathematics
Common Core State Standards -- Adopted June, 2010
School Year / Standards to be Taught / Standards to be Assessed2012-2013 / CCSS / CCSS (NC Testing)
2013-2014 / CCSS / CCSS (NC Testing)
2014-2015 / CCSS / CCSS (SBAC --- National Testing)
SMARTER Balanced Assessment Consortium
http://www.smarterbalanced.org/smarter-balanced-assessments/
Visit this website to see sample assessment items.
CCSS: Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
The Common Core State Standards for Mathematical Practice are expected to be integrated into every
mathematics lesson for all students, Grades K-12. Below are a few examples of how these Practices
may be integrated into tasks that students complete at grade 8.
Standards forMathematical Practice / Explanations and Examples
1. Make sense of problems & persevere in solving them. / In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves: “What is the most efficient way to solve the problem?”, “Does this make sense?”; and “Can I solve the problem in a different way?”
2. Reason abstractly and quantitatively. / In grade 8, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree of linearity of functions. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate the symbol ic representations by applying properties of operations.
3. Construct viable arguments and critique the reasoning of others. / In grade 8, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their thinking to others and respond to others’ thinking.
4. Model with mathematics. / In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students solve systems of linear equations and compare properties of functions provided in different forms. Students use scatterplots to represent data and describe associations between variables. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context.
5. Use appropriate tools strategically. / Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 8 may translate a set of data given in tabular form to a graphical representation to compare it to another data set. Students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal.
6. Attend to precision. / In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the number system, functions, geometric figures, and data displays.
7. Look for and make use of structure. / Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them in terms of congruence and similarity.
8. Look for and express regularity in repeated reasoning. / In grade 8, students use repeated reasoning to understand algorithms and make generalizations about patterns. Students use iterative processes to determine more precise rational approximations for irrational numbers. They analyze patterns of repeating decimals to identify the corresponding fraction. During multiple opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value. Students flexibly make connections between covariance, rates, and representations showing the relationships between quantities.
CCSS: Mathematics K – 8 Domains
Domains / K / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8Counting and Cardinality / CC
Operations and Algebraic Thinking / OA / 30-35% / 12-17% / 5-10%
Number and Operations in Base Ten / NBT / 5-10% / 22-27% / 22-27%
Measurement and Data / MD / 22-27% / 12-17% / 10-15%
Geometry / G / 10-15% / 12-17% / 2-7% / 17-22% / 25-30% / 20-25%
Number and Operations -- Fractions / NF / 20-25% / 27-32% / 47-52%
Ratios and Proportional Relationships / RP / 12-17% / 22-27%
The Number System / NS / 27-32% / 7-12% / 2-7%
Expressions and Equations / EE / 27-32% / 22-27% / 27-32%
Statistics and Probability / SP / 7-12% / 12-17% / 15-20%
Functions / F / 22-27%
The % ranges are weight distributions determined by NC DPI – Division of Accountability Services, 3-27-12.
Common Core State Standards (CCSS) for Mathematics
North Carolina Assessment Specifications Summary
READY EOG Assessments, Grades 3–8
Weight Distributions for Grades 3–5
Domain / Grade 3 / Grade 4 / Grade 5Operations and Algebraic Thinking / 30–35% / 12–17% / 5–10%
Number and Operations in Base Ten / 5–10% / 22–27% / 22–27%
Number and Operations—Fractions / 20–25% / 27–32% / 47–52%
Measurement and Data / 22–27% / 12–17% / 10–15%
Geometry / 10–15% / 12–17% / 2–7%
Total / 100% / 100% / 100%
Weight Distributions for Grades 6–8
Domain / Grade 6 / Grade 7 / Grade 8Ratios and Proportional Relationships / 12–17% / 22–27% / NA
The Number System / 27–32% / 7–12% / 2–7%
Expressions and Equations / 27–32% / 22–27% / 27–32%
Functions / NA / NA / 22–27%
Geometry / 12–17% / 22–27% / 20–25%
Statistics and Probability / 7–12% / 12–17% / 15–20%
Total / 100% / 100% / 100%
Purpose of the Assessments
● Edition 4 Grades 3–8 mathematics assessments and the Algebra I/Integrated I assessments will measure students’ proficiency on the Common Core State Standards (CCSS) for Mathematics, adopted by the North Carolina State Board of Education in June 2010.
● Assessment results will be used for school and district accountability under the READY Accountability Model and for Federal reporting purposes.
Curriculum Cycle
● June 2010: North Carolina State Board of Education adoption of the CCSS
● 2010–2011: Item development for the Next Generation of Assessments, Edition 4
● 2011–2012: Administration of stand-alone field tests of Edition 4 assessments
● 2012–2013: Operational administration of Edition 4 assessments aligned to the CCSS
Standards
● The CCSS is posted at: http://www.corestandards.org/
● Each grade includes a set of content standards.
● For high school, the CCSS groups the standards by conceptual categories rather than by grade or course. The CCSS suggests a course sequence for teaching standards by each pathway, traditional or integrated. However, the CCSS allows states to create their own sequence as long as the full set of standards is completed by the third year.
● North Carolina will teach and assess a common set of standards for the first-year high school course of mathematics. For the second and third high school years, schools or districts may follow either a traditional or integrated pathway.
● In addition to the content standards, the CCSS includes eight Standards for Mathematical Practice that cross domains, grade levels, and high school courses. Assessment items that are written for specific content standards will, as much as possible, also link to one or more of the mathematical practices.
Cognitive Rigor and Item Complexity
● Assessment items will be designed, developed, and classified to ensure that the cognitive rigor of the operational test forms align to the cognitive complexity and demands of the Common Core State Standards (CCSS) for Mathematics. These items will require students to not only recall information, but also apply concepts and skills and make decisions.
Types of Items
● The Grades 3 and 4 mathematics assessments will consist of four-response-option multiple-choice items.
● The Grades 5–8 mathematics assessments and the Algebra I/Integrated I assessment will consist of four-response-option multiple-choice items and a maximum of eight gridded-response items requiring numerical responses. NCEXTEND2 assessments will consist of three-response-option multiple-choice items and gridded-response items.
● All CCSS mathematics assessments will include both calculator-active and calculator-inactive sections. One-third to one-half of the Grades 3–8 assessments will be comprised of calculator-inactive items; approximately one-third of the High School assessments will be calculator inactive.
● The NCEXTEND1 mathematics alternate assessments will consist of fifteen performance-based, multiple-choice items.
Delivery Mode
● Grades 3–8 mathematics assessments will be designed for paper/pencil administrations and may have an online administration option.
● The Algebra I/Integrated I mathematics assessment will be designed for an online administration but will also be available in a paper/pencil version.
● NCEXTEND2 is an alternate assessment for students with disabilities whose IEP specifies an assessment aligned to the general content standards but based on modified academic achievement standards. The Grades 3–8 NCEXTEND2 mathematics assessments will closely follow the weightings of the standards on the general assessments. The NCEXTEND2 mathematics assessments will be designed for online administrations.
● NCEXTEND1 is an alternate assessment designed for students with significant cognitive disabilities whose IEP specifies an assessment aligned to the Extended Common Core State Standards (CCSS) and based on alternate academic achievement standards. The NCEXTEND1 mathematics assessments will be designed for paper/pencil administrations with online data entry by the assessor.