### 1: Number and Operations

#### 1.1: Understand, apply, and be computationally fluent with multiplication and division of fractions and decimals.

1.1.2: Understand and explain procedures for multiplying and dividing fractions by using the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division.

1.1.3: Understand and explain procedures for multiplying and dividing decimals by using the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number).

1.1.4: Use common procedures to multiply and divide fractions and decimals efficiently and accurately.

1.1.5: Convert from one unit to another in the metric system of measurement by using understanding of the relationships among the units and by multiplying and dividing decimals.

1.1.6: Convert from one unit to another in the customary system of measurement by using understanding of the relationships among the units and by multiplying and dividing fractions.

1.1.7: Multiply and divide fractions and decimals to solve problems, including multi-step problems.

#### 1.2: Understand, apply, and be computationally fluent with rational numbers, including negative numbers.

1.2.1: Understand negative numbers in terms of their position on the number line, their role in the system of all rational numbers, and in everyday situations (e.g., situations of owing money or measuring elevations above and below sea level).

1.2.2: Understand absolute value in terms of distance on the number line and simplify numerical expressions involving absolute value.

1.2.3: By applying properties of arithmetic and considering negative numbers in everyday contexts, explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense.

1.2.4: Understand positive integer exponents in terms of repeated multiplication and evaluate simple exponential expressions.

1.2.5: Effectively compute with and solve problems using rational numbers, including negative numbers.

#### 1.3: Understand, estimate, and represent real numbers, including common irrational numbers and with scientific notation.

1.3.4: Understand and determine the square roots of perfect squares.

1.3.5: Understand and estimate square roots of non-perfect-squares, and determine more precise values using a calculator.

1.3.6: Represent, use, and interpret numbers in scientific notation.

1.3.7: Use scientific notation and rational and irrational numbers to model and solve problems.

#### 1.4: Understand and apply ratio and rate, including percents, and connect ratio and rate to fractions and decimals.

1.4.1: Build on understanding of fractions and part-whole relationships to understand ratios (by, for example, analyzing the relative quantities of boys and girls in the classroom or triangles and squares in a drawing).

1.4.2: Understand percent as a rate and develop fluency in converting among fractions, decimals, and percents.

1.4.3: Understand equivalent ratios as deriving from, and extending, pairs of rows (or columns) in the multiplication table.

1.4.5: Use a variety of strategies to solve problems involving ratio and rate.

#### 1.5: Understand and apply proportional reasoning.

1.5.1: Understand that a proportion is an equation that states that two ratios are equivalent.

1.5.2: Understand proportional relationships (y = kx or y/x = k), and distinguish proportional relationships from other relationships, including inverse proportionality (xy = k or y = k/x).

1.5.3: Understand that in a proportional relationship of two variables, if one variable doubles or triples, for example, then the other variable also doubles or triples, and if one variable changes additively by a specific amount, a, then the other variable changes additively by the amount ka.

1.5.4: Graph proportional relationships and identify the constant of proportionality as the slope of the related line.

1.5.5: Use ratios and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.

### 2: Algebra

#### 2.1: Write, interpret, and use mathematical expressions and equations, find equivalent forms, and relate such symbolic representations to verbal, graphical, and tabular representations.

2.1.1: Write mathematical expressions, equations, and formulas that correspond to given situations.

2.1.2: Understand that variables represent numbers whose exact values are not yet specified, use single letters, words, or phrases as variables, and use variables appropriately.

2.1.3: Evaluate expressions (for example, find the value of 3x if x is 7).

2.1.5: Understand that solutions of an equation are the values of the variables that make the equation true.

2.1.6: Solve simple one-step equations (i.e., involving a single operation) by using number sense, properties of operation, and the idea of maintaining equality on both sides of an equation.

2.1.7: Construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and use equations to describe simple relationships shown in a table (such as 3x = y).

2.1.8: Use expressions, equations, and formulas to solve problems, and justify their solutions.

#### 2.2: Understand and apply proportionality.

2.2.1: Understand that a proportion is an equation that states that two ratios are equivalent.

2.2.2: Understand proportional relationships (y = kx or y/x = k), and distinguish proportional relationships from other relationships, including inverse proportionality (xy = k or y = k/x).

2.2.3: Graph proportional relationships and identify the constant of proportionality as the slope of the related line.

2.2.4: Use ratios and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.

#### 2.3: Understand, solve, and apply linear equations and inequalities.

2.3.1: Make strategic choices of procedures to solve linear equations and inequalities in one variable and implement them efficiently.

2.3.2: Recognize and generate equivalent forms of linear expressions, by using the associative, commutative, and distributive properties.

2.3.3: Understand that when properties of equality are used to transform an equation into a new equivalent equation, solutions obtained for the new equation also solve the original equation.

2.3.4: Solve more complicated linear equations, including solving for one variable in terms of another.

2.3.5: Solve linear inequalities and represent the solution on a number line.

2.3.6: Formulate linear equations and inequalities in one variable and use them to solve problems, including in applied settings, and justify the solution using multiple representations.

#### 2.4: Understand and apply linear functions.

2.4.1: Understand linear functions and slope of lines in terms of constant rate of change.

2.4.2: Understand that the slope of a line is constant, for example by using similar triangles (e.g., as shown in the rise and run of "slope triangles"), and compute the slope of a line using any two points on the line.

2.4.3: Build on the concept of proportion, recognizing a proportional relationship (y/x = k, or y = kx) as a special case of a linear function. In this special case, understand that if one variable doubles or triples, for example, then the other variable also doubles or triples; and understand that if the input, or x-coordinate in this case, changes additively by a specific amount, a, then the output, or y-coordinate in this case, changes additively by the amount ka.

2.4.4: Understand that the graph of the equation y = mx + b is a line with y-intercept b and slope m.

2.4.5: Translate among verbal, tabular, graphical, and algebraic representations of functions, including recursive representations such as NEXT = NOW +3 (recognizing that tabular and graphical representations often only yield approximate solutions), and describe how such aspects of a linear function as slope, constant rate of change, and intercepts appear in different representations.

2.4.6: Use linear functions, and understanding of the slope of a line and constant rate of change, to analyze situations and solve problems.

#### 2.5: Use tables and graphs to analyze systems of linear equations.

2.5.1: Use tables and graphs to analyze and (approximately) solve systems of two linear equations in two variables.

2.5.2: Relate a system of two linear equations in two variables to a pair of lines in the plane that intersect, are parallel, or are the same.

### 3: Geometry

#### 3.1: Understand, determine, and apply area of polygons.

3.1.2: Use knowledge of area of simpler shapes to help find area of more complex shapes.

3.1.3: Understand and apply formulas to find area of triangles and quadrilaterals.

#### 3.2: Understand and apply similarity, with connections to proportion.

3.2.1: Understand that two objects are similar if they have the same shape (i.e., corresponding angles are congruent) but not necessarily the same size.

3.2.2: Understand similarity in terms of a scale factor between corresponding lengths in similar objects (i.e., similar objects are related by transformations of magnifying or shrinking).

3.2.3: Understand that relationships of lengths within similar objects are preserved (i.e., ratios of corresponding sides in similar objects are equal).

3.2.4: Understand that congruent figures are similar with a scale factor of 1.

3.2.5: Use understanding of similarity to solve problems in a variety of contexts.

#### 3.3: Understand, determine, and apply surface area and volume of prisms and cylinders and circumference and area of circles.

3.3.1: Find the area of more complex two-dimensional shapes, such as pentagons, hexagons, or irregular shaped regions, by decomposing the complex shapes into simpler shapes, such as triangles.

3.3.2: Understand that the ratio of the circumference to the diameter of a circle is constant and equal to pi, and use this fact to develop a formula for the circumference of a circle.

3.3.3: Understand that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram.

3.3.4: Develop and justify strategies for determining the surface area of prisms and cylinders by determining the areas of shapes that comprise the surface.

3.3.5: By decomposing prisms and cylinders by slicing them, develop and understand formulas for their volumes (Volume = Area of base x Height).

3.3.6: Select appropriate two-and three-dimensional shapes to model real-world situations and solve a variety of problems (including multi-step problems) involving surface area, area and circumference of circles, and volume of prisms and cylinders.

#### 3.4: Analyze two-dimensional space and figures by using distance, angle, coordinates, and transformations.

3.4.1: Explore and explain the relationships among angles when a transversal cuts parallel lines.

3.4.2: Use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and apply this fact about triangles to find unknown measures of angles.

3.4.3: Understand and explain how particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines (e.g., "slope triangles").

3.4.4: Use reasoning about similar triangles to solve a variety of problems, including those that involve determining heights and distances.

3.4.5: Explain why the Pythagorean Theorem is valid by using a variety of methods ? for example, by decomposing a square in different ways.

3.4.6: Apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane and to measure lengths and analyze polygons.

3.4.7: Understand and apply transformations ? reflection, translation, rotation, and dilation, and understand similarity, congruence, and symmetry in terms of transformations.

#### 3.5: Visualize, represent, and describe three-dimensional shapes.

3.5.1: Recognize and draw two-dimensional representations of three-dimensional figures, including nets, front-side-top views, and perspective drawings.

3.5.2: Identify and describe three-dimensional shapes, including prisms, pyramids, cylinders, and spheres.

3.5.3: Examine, build, compose, and decompose three-dimensional objects, using a variety of tools, including paper-and-pencil, geometric models, and dynamic geometry software.

3.5.4: Use visualization and three-dimensional shapes to solve problems, especially in real-world settings.

### 4: Data Analysis and Probability

#### 4.1: Understand, interpret, determine, and apply measures of center and graphical representations of data.

4.1.1: Extend prior work with mode, median, and mean as measures of center.

4.1.2: Compute the mean for small data sets and explore its meaning as a balance point for a data set.

4.1.3: Extend prior work with bar graphs, line graphs, line plots, histograms, circle graphs, and stem and leaf plots as graphical representations of data to include box-and-whisker plots and scatterplots.

4.1.4: Create and interpret box-and-whisker plots and scatterplots.

#### 4.2: Analyze and summarize data sets, including initial analysis of variability.

4.2.1: Select, determine, explain, and interpret appropriate measures of center for given data sets (mean, median, mode).

4.2.2: Select, create, explain, and interpret appropriate graphical representations for given data sets (bar graphs, circle graphs, line graphs, histograms, line plots, stem and leaf plots, box-and-whisker plots, scatterplots).

4.2.3: Summarize and compare data sets using appropriate numerical statistics and graphical representations.

4.2.4: Compare the information provided by the mean and the median and investigate the different effects that changes in the data values have on these measures of center.

4.2.5: Understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center, and thus consider and describe the variability of the data (e.g., range and interquartile range).

4.2.6: Informally determine a line of best fit for a scatterplot to make predictions and estimates.

4.2.7: Formulate questions, gather data relevant to the questions, organize and analyze the data to help answer the questions, including informal analysis of randomness and bias.

#### 4.3: Use proportions and percentages to analyze data and chance.

4.3.1: Use proportions to make estimates relating to a population on the basis of a sample.

4.3.2: Apply percentages to make and interpret histograms and circle graphs.

4.3.3: Explore situations in which all outcomes of an experiment are equally likely, and thus the theoretical probability of an event is the number of outcomes corresponding to the event divided by total number of possible outcomes.

4.3.4: Use theoretical probability and proportions to make approximate predictions.

#### 4.4: Understand and represent simple probabilistic situations.

4.4.1: Represent the probability of events that are impossible, unlikely, likely, and certain using rational numbers from 0 to 1.

4.4.2: List all possible outcomes of a given experiment or event.

#### 4.5: Understand, compute, and estimate simple probabilities using counting strategies and simulation.

4.5.2: Compute probabilities for compound events, using such methods as organized lists, tree diagrams (counting trees), area models, and counting principles.

4.5.3: Estimate the probability of simple and compound events through experimentation and simulation.

4.5.4: Use a variety of experiments to explore the relationship between experimental and theoretical probabilities and the effect of sample size on this relationship.

Correlation last revised: 1/20/2017

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.