Mathematics Curriculum Framework
November 2000

Learning Standards

MCAS Question Search
MCAS Open Response Questions

Number Sense and Operations
for Grades 7-8

MCAS Number Sense and Operations Questions

cross symbol Understand numbers, ways of representing numbers, relationships among numbers, and number systems
 
solid circle symbol Understand meanings of operations and how they relate to one another
 
solid triangle symbol Compute fluently and make reasonable estimates

Learning Standards

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

8.N.1 Compare, order, estimate, and translate among integers, fractions and mixed numbers (i.e., rational numbers), decimals, and percents.
  Visual Fractions
Grampy Game
Comparing Fractions
two circles, two fractions
(enlvm)
 Multiplying Mixed Numbers
(math.com)
   
8.N.2 Define, compare, order, and apply frequently used irrational numbers, such as √2 and π.
8.N.3 Use ratios and proportions in the solution of problems, in particular, problems involving unit rates, scale factors, and rate of change.
8.N.4 Represent numbers in scientific notation, and use them in calculations and problem situations.
8.N.5 Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.
8.N.6 Demonstrate an understanding of absolute value, e.g., |-3| = |3| = 3.
8.N.7 Apply the rules of powers and roots to the solution of problems. Extend the Order of Operations to include positive integer exponents and square roots.
8.N.8 Demonstrate an understanding of the properties of arithmetic operations on rational numbers. Use the associative, commutative, and distributive properties; properties of the identity and inverse elements (e.g., -7 + 7 = 0; 3/4 x 4/3 = 1); and the notion of closure of a subset of the rational numbers under an operation (e.g., the set of odd integers is closed under multiplication but not under addition).
8.N.9 Use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems, e.g. multiplying by 1/2 or 0.5 is the same as dividing by 2.
8.N.10 Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1).
8.N.11 Determine when an estimate rather than an exact answer is appropriate and apply in problem situations.
8.N.12 Select and use appropriate operations—addition, subtraction, multiplication, division, and positive integer exponents—to solve problems with rational numbers (including negatives).

Patterns, Relations, and Algebra
for Grades 7-8

MCAS Patterns, Relations, and Algebra Questions

cross symbol Understand patterns, relations, and functions
solid circle symbol Represent and analyze mathematical situations and structures using algebraic symbols
solid triangle symbol Use mathematical models to represent and understand quantitative relationships
solid square symbol Analyze change in various contexts

Learning Standards

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

8.P.1 Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric progressions, e.g., compounding.
   
8.P.2 Evaluate simple algebraic expressions for given variable values, e.g., 3a2 - b for a = 3 and b = 7.
 

Understanding Algebra
(Math is Fun)

Balancing Equation
(maths net)

NCTM - Algebraic Balance

 
8.P.3 Demonstrate an understanding of the identity (-x)(-y) = xy. Use this identity to simplify algebraic expressions, e.g., (-2)(-x+2) = 2x - 4.
8.P.4 Create and use symbolic expressions and relate them to verbal, tabular, and graphical representations.
8.P.5 Identify the slope of a line as a measure of its steepness and as a constant rate of change from its table of values, equation, or graph. Apply the concept of slope to the solution of problems.
8.P.6 Identify the roles of variables within an equation, e.g., y = mx + b, expressing y as a function of x with parameters m and b.
8.P.7 Set up and solve linear equations and inequalities with one or two variables, using algebraic methods, models, and/or graphs.
   
8.P.8 Explain and analyze-both quantitatively and qualitatively, using pictures, graphs, charts, or equations-how a change in one variable results in a change in another variable in functional relationships, e.g., C = πd, A = πr2 (A as a function of r), Arectangle = lw (Arectangle as a function of l and w).
8.P.9 Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate.
8.P.10 Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and x- and y-intercepts of different linear patterns.

Geometry
for Grades 7-8

MCAS Geometry Questions

cross symbol Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
solid circle symbol Specify locations and describe spatial relationships using coordinate geometry and other representational systems
solid triangle symbol Apply transformations and use symmetry to analyze mathematical situations
solid square symbol Use visualization, spatial reasoning, and geometric modeling to solve problems

Learning Standards

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

8.G.1 Analyze, apply, and explain the relationship between the number of sides and the sums of the interior and exterior angle measures of polygons.
8.G.2 Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems.
8.G.3 Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines cut by a transversal.
8.G.4 Demonstrate an understanding of the Pythagorean theorem. Apply the theorem to the solution of problems.
8.G.5 Use a straight-edge, compass, or other tools to formulate and test conjectures, and to draw geometric figures.
8.G.6 Predict the results of transformations on unmarked or coordinate planes and draw the transformed figure, e.g., predict how tessellations transform under translations, reflections, and rotations.
8.G.7 Identify three-dimensional figures (e.g., prisms, pyramids) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces.
8.G.8 Recognize and draw two-dimensional representations of three-dimensional objects, e.g., nets, projections, and perspective drawings.

Measurement
for Grades 7-8

MCAS Measurement Questions

cross symbol Understand measurable attributes of objects and the units, systems, and processes of measurement
solid circle symbol Apply appropriate techniques, tools, and formulas to determine measurements

Learning Standards

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

8.M.1 Select, convert (within the same system of measurement), and use appropriate units of measurement or scale.
8.M.2 Given the formulas, convert from one system of measurement to another. Use technology as appropriate.
8.M.3 Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
8.M.4 Use ratio and proportion (including scale factors) in the solution of problems, including problems involving similar plane figures and indirect measurement.
8.M.5 Use models, graphs, and formulas to solve simple problems involving rates, e.g., velocity and density.

 

Data Analysis, Statistics, and Probability
for Grades 7-8

MCAS Data Analysis, Statistics, and Probability Questions

cross symbol Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
solid circle symbol Select and use appropriate statistical methods to analyze data
solid triangle symbol Develop and evaluate inferences and predictions that are based on data
solid square symbol Understand and apply basic concepts of probability

Learning Standards

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:

8.D.1 Describe the characteristics and limitations of a data sample. Identify different ways of selecting a sample, e.g., convenience sampling, responses to a survey, random sampling.
8.D.2 Select, create, interpret, and utilize various tabular and graphical representations of data, e.g., circle graphs, Venn diagrams, scatterplots, stem-and-leaf plots, box-and-whisker plots, histograms, tables, and charts. Differentiate between continuous and discrete data and ways to represent them.
8.D.3 Find, describe, and interpret appropriate measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data. Use these notions to compare different sets of data.
8.D.4 Use tree diagrams, tables, organized lists, basic combinatorics ("fundamental counting principle"), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.