 # Math Standards

I.   The Number System

7.   NS.1- Extend prior knowledge of operations with positive rational numbers to add and to subtract all rational numbers and represent the sum or difference on a number line.

a.  Understand that the additive inverse of a number is its opposite and their sum is equal to zero.

b. Understand that the sum of two rational numbers (𝑝+𝑞) represents a distance from p on the number line equal to

|q| where the direction is indicated by the sign of q.

c.  Translate between the subtraction of rational numbers and addition using the additive inverse, 𝑝 − 𝑞 = 𝑝 + (−𝑞).

d. Demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference.

e.Apply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to add and subtract rational numbers.

7. NS.2-Extend prior knowledge of operations with positive rational numbers to multiply and to divide all rational numbers.

a.  Understand that the multiplicative inverse of a number is its reciprocal and their product is equal to one.

b. Understand sign rules for multiplying rational numbers.

c.  Understand sign rules for dividing rational numbers and that a quotient of integers (with a non-zero divisor) is a rational number.

d. Apply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to multiply and divide rational numbers.

e.  Understand that some rational numbers can be written as integers and all rational numbers can be written as fractions or decimal numbers that terminate or repeat.

7.NS.3- Apply the concepts of all four operations with rational numbers to solve real-world and mathematical problems. 7.NS.4- Understand and apply the concepts of comparing and ordering to rational numbers.

a.Interpret statements using less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), and equal to (=) as relative locations on the number line.

b. Use concepts of equality and inequality to write and explain real-world and mathematical situations.

7.NS.5-Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, percentages). Exclude the conversion of repeating decimal numbers to fractions.

II.   Geometry and Measurement

7.GM.1- Determine the scale factor and translate between scale models and actual measurements (e.g., lengths, area) of real- world objects and geometric figures using proportional reasoning.

7. GM.2-Construct triangles and special quadrilaterals using a variety of tools (e.g., freehand, ruler and protractor, technology).

a.  Construct triangles given all measurements of either angles or sides.

b. Decide if the measurements determine a unique triangle, more than one triangle, or no triangle.

c.  Construct special quadrilaterals (i.e., kite, trapezoid, isosceles trapezoid, rhombus, parallelogram, rectangle) given specific parameters about angles or sides.

7.GM.3 –Describe two-dimensional cross-sections of three-dimensional figures, specifically right rectangular prisms and right rectangular pyramids.

7. GM.4- Investigate the concept of circles.

a.  Demonstrate an understanding of the proportional relationships between diameter, radius, and circumference of a circle.

b. Understand that the constant of proportionality between the circumference and diameter is

c.  Explore the relationship between circumference and area using a visual model.

d. Use the formulas for circumference and area of circles appropriately to solve real-world and mathematical problems.

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7. GM.6-Apply the concepts of two- and three-dimensional figures to real-world and mathematical situations.

a.  Understand that the concept of area is applied to two-dimensional figures such as triangles, quadrilaterals, and polygons.

b. Understand that the concepts of volume and surface area are applied to three-dimensional figures such as cubes, right rectangular prisms, and right triangular prisms.

c.  Decompose cubes, right rectangular prisms, and right triangular prisms into rectangles and triangles to derive the formulas for volume and surface area.

d. Use the formulas for area, volume, and surface area appropriately.

III.    Data Analysis, Statistics, and Probability

7. DSP.1- Investigate concepts of random sampling.

a.  Understand that a sample is a subset of a population and both possess the same characteristics.

b. Differentiate between random and non-random sampling.

c.  Understand that generalizations from a sample are valid only if the sample is representative of the population.

d. Understand that random sampling is used to gather a representative sample and supports valid inferences about the population.

7.DSP.2 -Draw inferences about a population by collecting multiple random samples of the same size to investigate variability in estimates of the characteristic of interest.

7.DSP.3-Visually compare the centers, spreads, and overlap of two displays of data (i.e. dot plots, histograms, box plots) that are graphed on the same scale and draw inferences about this data.

7.DSP.4 –Compare the numerical measures of center (mean, median, mode) and variability (range, interquartile range, mean absolute deviation) from two random samples to draw inferences about the populations.

7. DSP.5- Investigate the concept of probability of chance events.

a.  Determine probabilities of simple events.

b. Understand that probability measures likelihood of a chance event occurring.

c.  Understand that the probability of a chance event is a number between 0 and 1.

d. Understand that a probability closer to 1 indicates a likely chance event.

e.  Understand that a probability close to 12 indicates that a chance event is neither likely nor unlikely.

f. Understand that a probability closer to 0 indicates an unlikely chance event.

7. DSP.6-Investigate the relationship between theoretical and experimental probabilities for simple events.

a.  Determine approximate outcomes using theoretical probability.

b. Perform experiments that model theoretical probability.

c.  Compare theoretical and experimental probabilities.

7. DSP.7- Apply the concepts of theoretical and experimental probabilities for simple events.

a.  Differentiate between uniform and non-uniform probability models (distributions).

b. Develop both uniform and non-uniform probability models.

c.  Perform experiments to test the validity of probability models.

7. DSP.8-Extend the concepts of simple events to investigate compound events.

a.  Understand that the probability of a compound event is between 0 and 1.

b. Identify the outcomes in a sample space using organized lists, tables, and tree diagrams.

c.  Determine probabilities of compound events using organized lists, tables, and tree diagrams.

d. Design and use simulations to collect data and determine probabilities.

e.  Compare theoretical and experimental probabilities for compound events.

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A mathematically literate student can:

1.  Make sense of problems and persevere in solving them.

a.   Relate a problem to prior knowledge.

b.   Recognize there may be multiple entry points to a problem and more than one path to a solution.

c.   Analyze what is given, what is not given, what is being asked, and what strategies are needed, and make an initial attempt to solve a problem.

d.   Evaluate the success of an approach to solve a problem and refine it if necessary.

2.  Reason both contextually and abstractly.

a.   Make sense of quantities and their relationships in mathematical and real-world situations.

b.   Describe a given situation using multiple mathematical representations.

c.   Translate among multiple mathematical representations and compare the meanings each representation conveys about the situation.

d.   Connect the meaning of mathematical operations to the context of a given situation.

I. The Number System

8. NS.1-Explore the real number system and its appropriate usage in real-world situations.

a.   Recognize the differences between rational and irrational numbers.

b.   Understand that all real numbers have a decimal expansion.

c.   Model the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers.

8.NS.2 –Estimate and compare the value of irrational numbers by plotting them on a number line.

8.NS.3-Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, percentages). Include the conversion of repeating decimal numbers to fractions.

II.   Functions

8.F.1-Explore the concept of functions.

a.   Understand that a function assigns to each input exactly one output.

b.   Relate inputs (𝑥-values or domain) and outputs (𝑦-values or range) to independent and dependent variables.

c.   Translate among the multiple representations of a function, including mappings, tables, graphs, equations, and verbal descriptions.

d.   Determine if a relation is a function using multiple representations, including mappings, tables, graphs, equations, and verbal descriptions.

e.   Graph a function from a table of values. Understand that the graph and table both represent a set of ordered pairs of that function.

8.F.2-Compare multiple representations of two functions, including mappings, tables, graphs, equations, and verbal descriptions, in order to draw conclusions.

8.F.3-Investigate the differences between linear and nonlinear functions using multiple representations (i.e. tables, graphs, equations, and verbal descriptions).

a.   Define an equation in slope-intercept form (𝑦=𝑚𝑥+𝑏) as being a linear  function.

b.   Recognize that the graph of a linear function has a constant rate of change.

c.   Provide examples of nonlinear functions.

8.F.4     –Apply the concepts of linear functions to real-world and mathematical situations.

a.   Understand that the slope is the constant rate of change and the 𝑦-intercept is the point where 𝑥 = 0.

b.   Determine the slope and the 𝑦-intercept of a linear function given multiple representations, including two points, tables, graphs, equations, and verbal descriptions.

c.   Construct a function in slope-intercept form that models a linear relationship between two quantities.

d.   Interpret the meaning of the slope and the 𝑦-intercept of a linear function in the context of the situation.

e.   Explore the relationship between linear functions and arithmetic sequences.

8.F.5     –Apply the concepts of linear and nonlinear functions to graphs in real-world and mathematical situations.

a.   Analyze and describe attributes of graphs of functions (e.g., constant, increasing/decreasing, linear/nonlinear, maximum/minimum, discrete/continuous).

b.   Sketch the graph of a function from a verbal description.

c.   Write a verbal description from the graph of a function with and without scales.

III.      Expressions, Equations, and Inequalities

8.EEI.1- Understand and apply the laws of exponents (i.e. product rule, quotient rule, power to a power, product to a power, quotient to a power, zero power property, negative exponents) to simplify numerical expressions that include integer exponents

8. EEI.2- Investigate concepts of square and cube roots.

a.   Find the exact and approximate solutions to equations of the form 𝑥2=𝑝 and 𝑥3=𝑝 where 𝑝 is a positive rational number.

b.   Evaluate square roots of perfect squares.

c.   Evaluate cube roots of perfect cubes.

d.   Recognize that square roots of non-perfect squares are irrational.

8. EEI.3- Explore the relationship between quantities in decimal and scientific notation.

a.   Express very large and very small quantities in scientific notation in the form 𝑎×10𝑏=𝑝 where 1≤𝑎<10 and 𝑏 is an integer.

b.   Translate between decimal notation and scientific notation.

c.   Estimate and compare the relative size of two quantities in scientific notation.

8. EEI.4-Apply the concepts of decimal and scientific notation to solve real-world and mathematical problems.

a.   Multiply and divide numbers expressed in both decimal and scientific notation.

b.   Select appropriate units of measure when representing answers in scientific notation.

c.   Translate how different technological devices display numbers in scientific notation.

8. EEI.5 –Apply concepts of proportional relationships to real-world and mathematical situations.

a.  Graph proportional relationships.

b.   Interpret unit rate as the slope of the graph.

c.  Compare two different proportional relationships given multiple representations, including tables, graphs, equations, diagrams, and verbal descriptions.

8. EEI.6 –Apply concepts of slope and 𝑦-intercept to graphs, equations, and proportional relationships.

a.   Explain why the slope, 𝑚, is the same between any two distinct points on a non-vertical line using similar triangles.

b.   Derive the slope-intercept form (𝑦=𝑚𝑥+𝑏) for a non-vertical   line.

c.   Relate equations for proportional relationships (𝑦=𝑘𝑥)  with the  slope-intercept form (𝑦=𝑚𝑥+𝑏)     where

𝑏=0.

8.EEI.7 –Extend concepts of linear equations and inequalities in one variable to more complex multi-step equations and inequalities in real-world and mathematical situations.

a.   Solve linear equations and inequalities with rational number coefficients that include the use of the distributive property, combining like terms, and variables on both sides.

b.   Recognize the three types of solutions to linear equations: one solution (𝑥=𝑎), infinitely many solutions (𝑎=𝑎), or no solutions    (𝑎=𝑏).

c.   Generate linear equations with the three types of solutions.

d.   Justify why linear equations have a specific type of solution

8.E.E.1.8-Investigate and solve real-world and mathematical problems involving systems of linear equations in two variables with integer coefficients and solutions.

a.   Graph systems of linear equations and estimate their point of intersection.

b.   Understand and verify that a solution to a system of linear equations is represented on a graph as the point of intersection of the two lines.

c.   Solve systems of linear equations algebraically, including methods of substitution and elimination, or through inspection.

d.   Understand that systems of linear equations can have one solution, no solution, or infinitely many solutions.

IV.   Geometry and Measurement

8. GM.1- Investigate the properties of rigid transformations (rotations, reflections, translations) using a variety of tools (e.g., grid paper, reflective devices, graphing paper, technology).

a.   Verify that lines are mapped to lines, including parallel lines.

b.   Verify that corresponding angles are congruent.

c.   Verify that corresponding line segments are congruent.

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8. GM.2 -Apply the properties of rigid transformations (rotations, reflections, translations).

a.   Rotate geometric figures 90, 180, and 270 degrees, both clockwise and counterclockwise, about the origin.

b.   Reflect geometric figures with respect to the 𝑥-axis and/or 𝑦-axis.

c.   Translate geometric figures vertically and/or horizontally.

d.   Recognize that two-dimensional figures are only congruent if a series of rigid transformations can be performed to map the pre- image to the image.

e.   Given two congruent figures, describe the series of rigid transformations that justifies this congruence.

8. GM.3-Investigate the properties of transformations (rotations, reflections, translations, dilations) using a variety of tools (e.g., grid paper, reflective devices, graphing paper, dynamic software).

a.   Use coordinate geometry to describe the effect of transformations on two-dimensional figures.

b.   Relate scale drawings to dilations of geometric figures.

8. GM.4 -Apply the properties of transformations (rotations, reflections, translations, dilations).

a.   Dilate geometric figures using scale factors that are positive rational numbers.

b.   Recognize that two-dimensional figures are only similar if a series of transformations can be performed to map the pre-image to the image.

c.   Given two similar figures, describe the series of transformations that justifies this similarity.

d.   Use proportional reasoning to find the missing side lengths of two similar figures.

8. GM.5-Extend and apply previous knowledge of angles to properties of triangles, similar figures, and parallel lines cut by a transversal

a.   Discover that the sum of the three angles in a triangle is 180 degrees.

b.   Discover and use the relationship between interior and exterior angles of a triangle.

c.   Identify congruent and supplementary pairs of angles when two parallel lines are cut by a transversal.

d.   Recognize that two similar figures have congruent corresponding angles.

8.GM.6 -Use models to demonstrate a proof of the Pythagorean Theorem and its converse

8.GM.7 -Apply the Pythagorean Theorem to model and solve real-world and mathematical problems in two and three dimensions involving right triangles.

8.GM.8-Find the distance between any two points in the coordinate plane using the Pythagorean Theorem.

8.GM.9 -Solve real-world and mathematical problems involving volumes of cones, cylinders, and spheres and the surface area of cylinders.

V.  Data Analysis, Statistics, and Probability

8. DSP.1-Investigate bivariate data.

a.  Collect bivariate data.

b.   Graph the bivariate data on a scatter plot.

c.   Describe patterns observed on a scatter plot, including clustering, outliers, & association (+, -, no correlation, linear, nonlinear).

8.DSP.2 Draw an approximate line of best fit on a scatter plot that appears to have a linear association and informally assess the fit of the line to the data points.

8. DSP.3 Apply concepts of an approximate line of best fit in real-world situations.

a.   Find an approximate equation for the line of best fit using two appropriate data points.

b.   Interpret the slope and intercept.

c.   Solve problems using the equation.

8.DSP.4* Investigate bivariate categorical data in two-way tables.

d.   Organize bivariate categorical data in a two-way table.

e.   Interpret data in two-way tables using relative frequencies.

f.   Explore patterns of possible association between the two categorical variables.

8.DSP.5* Organize data in matrices with rational numbers and apply to real-world and mathematical situations

g.   Understand that a matrix is a way to organize data.

h.   Recognize that a 𝑚×𝑛 matrix has 𝑚 rows and 𝑛 columns.

i.  Add and subtract matrices of the same size.

Multiply a matrix by a scalar.