NCTM Standards for Grades 6-8

6-8. NO : Number and Operations

Understand numbers, ways of representing numbers, relationships among numbers, and number systems Understand numbers, ways of representing numbers, relationships among numbers, and number systems

6-8. NO.1.4
Understand and use ratios and proportions to represent quantitative relationships.
What is a Ratio Rates and Ratios Writing Ratios and Solving Proportions Ratios
6-8. NO.1.6
Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Prime Numbers Factors Prime Factorization I Prime Factorization II Understanding GCF and LCM Greatest Common Factor and Least Common Multiple
6-8. NO.1.7
Develop meaning for integers and represent and compare quantities with them.
Using Integers to Represent Situations Understanding Integers: One Greater Working with Integers

Understand meanings of operations and how they relate to one another

6-8. NO.2.2
Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.
Number Properties Using Number Properties to Calculate Mentally
6-8. NO.2.3
Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
Dividing Fractions (Using Patterns with Multiplication) Integer Subtraction by Adding the Opposite Squares and Square Roots

Compute fluently and make reasonable estimates

6-8. NO.3.2
Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.
Adding Fractions Using Models Solving Problems with Integers I Multiplying Fractions Using Models Multiplying Mixed Numbers Add, Subtract, Multiply, or Divide Integers Adding and Subtracting Like and Unlike Fractions Adding Fractions Adding Mixed Numbers Subtracting Fractions Subtracting Mixed Numbers Adding and Subtracting Like and Unlike Fractions and Mixed Numbers Multiplying Mixed and Whole Numbers I Multiplying Mixed and Whole Numbers II Dividing Fractions Dividing Mixed Numbers Addition and Subtraction of Decimals to the Tenths Place with Numbers Less than Ten Addition and Subtraction of Decimals to the Tenths Place with Numbers Less than Twenty Addition and Subtraction of Decimals to the Hundredths Place with Numbers Less than Ten Addition and Subtraction of Decimals to the Hundredths Place with Numbers Less than Twenty Addition and Subtraction of Decimals to the Tenths Place with Numbers Greater than Ten Addition and Subtraction of Decimals to the Hundredths Place with Numbers Greater than Twenty Addition and Subtraction of Decimals (Including Decimals Presented as Written Words) Addition and Subtraction of Decimals (Including More Than Two Addends) Multiplication of a Whole Number by a Decimal (Tenth and Hundredth) Multiplication of a Decimal (Tenth) by a Decimal (Tenth and Hundredth) Multiplication of Decimals More Multiplication of Decimals Review of Multiplication with Decimals More Review of Multiplication with Decimals Division of Whole Number of Decimal (Tenth) and a Decimal Division of Decimals Working with Integers Adding Integers With and Without a Number Line Using Integer Addition and Subtraction Concepts to Solve Problems I Using Integer Addition and Subtraction Concepts to Solve Problems III Solving Problems with Integers II
6-8. NO.3.3
Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
Rounding and Estimating Estimating Square Roots
6-8. NO.3.4
Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
Using Proportions to Solve Problems Scale Factors Writing Ratios and Solving Proportions Using Proportions Percent Proportion

6-8. A : Algebra

Understand patterns, relations, and functions

6-8. A.1.2
Relate and compare different forms of representation for a relationship.
Equation of a Line in y = mx + b Form
6-8. A.1.3
Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

Represent and analyze mathematical situations and structures using algebraic symbols Represent and analyze mathematical situations and structures using algebraic symbols

6-8. A.2.1
Develop an initial conceptual understanding of different uses of variables.
Evaluating Decimal Expressions and Functions Input/Output Tables
6-8. A.2.2
Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
Calculating Slope Equation of a Line in y = mx + b Form Positive and Negative Slope

Use mathematical models to represent and understand quantitative relationships Use mathematical models to represent and understand quantitative relationships

6-8. A.3.1
Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
Writing More Algebraic Expressions: Representing Quantities in Real World Situations

Analyze change in various contexts

6-8. A.4.1
Use graphs to analyze the nature of changes in quantities in linear relationships.

6-8. G : Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

6-8. G.1.1
Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
Angle Measures: Triangles and Parallel Lines Cut by a Transversal Types of Triangles Angle Measures Properties of Quadrilaterals More Properties of Quadrilaterals Medians and Bisectors Angle Measures of Regular Polygons
6-8. G.1.2
Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
Similar Figures Finding Missing Measures
6-8. G.1.3
Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
Similar Figures Finding Missing Measures Pythagorean Relationships

Specify locations and describe spatial relationships using coordinate geometry and other representational systems Specify locations and describe spatial relationships using coordinate geometry and other representational systems

6-8. G.2.1
Use coordinate geometry to represent and examine the properties of geometric shapes.
Distance Between Points on a Coordinate Plane
6-8. G.2.2
Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.

Apply transformations and use symmetry to analyze mathematical situations

6-8. G.3.1
Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
Translations on a Coordinate Grid More Translations on a Coordinate Grid Scale Factors Reflections on a Coordinate Grid Reflections and Rotations on a Coordinate Grid
6-8. G.3.2
Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
Diagonals and Lines of Symmetry in Regular Polygons

Use visualization, spatial reasoning, and geometric modeling to solve problems

6-8. G.4.1
Draw geometric objects with specified properties, such as side lengths or angle measures.
Types of Triangles Properties of Quadrilaterals More Properties of Quadrilaterals Medians and Bisectors
6-8. G.4.2
Use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume.
6-8. G.4.3
Use visual tools such as networks to represent and solve problems.
6-8. G.4.4
Use geometric models to represent and explain numerical and algebraic relationships.
6-8. G.4.5
Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

6-8. M : Measurement

Understand measurable attributes of objects and the units, systems, and processes of measurement

6-8. M.1.1
Understand both metric and customary systems of measurement.
Convert Customary Units of Length Convert Metric Units of Length, Mass and Capacity
6-8. M.1.2
Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

Apply appropriate techniques, tools, and formulas to determine measurements

6-8. M.2.1
Use common benchmarks to select appropriate methods for estimating measurements.
Volume of a Cylinder Volume of a Prism Volume of a Pyramid
6-8. M.2.4
Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.
Surface Area of Prisms Volume of a Cylinder Volume of a Prism Surface Area of Cylinders Volume of a Pyramid
6-8. M.2.5
Solve problems involving scale factors, using ratio and proportion.
Scale Factors
6-8. M.2.6
Solve simple problems involving rates and derived measurements for such attributes as velocity and density.

6-8. DAP : Data Analysis and Probability

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

6-8. DAP.1.1
Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.
6-8. DAP.1.2
Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.
Circle Graphs

Select and use appropriate statistical methods to analyze data

6-8. DAP.2.1
Find, use, and interpret measures of center and spread, including mean and interquartile range.
Median and Mode More Range, Minimum, Maximum and Mean Range, Minimum, Maximum and Mean
6-8. DAP.2.2
Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

Develop and evaluate inferences and predictions that are based on data

6-8. DAP.3.1
Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken.
6-8. DAP.3.2
Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.
6-8. DAP.3.3
Use conjectures to formulate new questions and plan new studies to answer them.

Understand and apply basic concepts of probability Understand and apply basic concepts of probability

6-8. DAP.4.1
Understand and use appropriate terminology to describe complementary and mutually exclusive events.
Complementary and Mutually Exclusive Events
6-8. DAP.4.2
Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.
6-8. DAP.4.3
Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
Counting Outcomes Permutations and Combinations III Permutations and Combinations II Permutations and Combinations Finding Probability Outcomes and Probability