# Middle School Course Acceleration

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4 Work with radicals and integer exponents. 8.3 Know and apply the properties of integer exponents to generate equivalent numerical expressions. [8-EE1] 8.4 Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. [8-EE2] 8.5 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. [8-EE3] 8.6 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities[8-ee4] Clusters with Instructional Notes Analyze proportional relationships and use them to solve real-world and mathematical problems. Unit 2: Proportionality & Linear Relationships 7.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. [7-RP1] 7.2 Recognize and represent proportional relationships between quantities. [7- RP2] a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.[7-rp2a] b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. [7-RP2b] c. Represent proportional relationships by equations. [7-RP2c] d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. [7-RP2d] Use properties of operations to generate equivalent expressions. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.3 Use proportional relationships to solve multistep ratio and percent problems. [7-RP3] 7.7 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. [7-EE1] 7.8 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. [7-EE2] 7.9 Solve multistep real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies. [7-EE3] 7.10 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. [7-EE4] a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. [7-EE4a] b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and in c. Interpret it in the context of the problem. [7-EE4b]

5 Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5] 8.8. Use similar triangles to explain why the slope m is the same between any two distinct point on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. [8-EE6] 8.9. Solve linear equations in one variable. [8-EE7] a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). [8-EE7a] b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. [8-EE7b] Clusters with Instructional Notes Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models. Unit 3: Introduction to Sampling and Inference Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7-SP1] Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7-SP2] Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. [7-SP3] Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. [7-SP4] Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7-SP5] Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. [7-SP6] Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7-SP7] a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. [7-SP7a] b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. [7-SP7b] Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7-SP8] a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. [7-SP8a] b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. [7-SP8b] c. Design and use a simulation to generate frequencies for compound events. [7-SP8c]

6 Clusters with Instructional Notes Draw, construct, and describe geometrical figures and describe the relationships between them. Unit 4: Creating, Comparing, and Analyzing Geometric Figures Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. [7-G1] Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. [7-G2] Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Understand congruence and similarity using physical models, transparencies, or geometry software. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. [7-G3] Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. [7-G4] 7.15 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. [7-G5] 7.16 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.[7-g6] Verify experimentally the properties of rotations, reflections, and translations: [8-G1] a. Lines are taken to lines, and line segment to line segments of the same length. [8-G1a] b. Angles are taken to angles of the same measure. [8-G1b] c. Parallel lines are taken to parallel lines. [8-G1c] Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8-G2] Describe the effect of dilations, translations, rotations and reflections on twodimensional figures using coordinates. [8-G3] Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. [8-G4] 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. [8-G5] Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. [8-G9]

8 Below you will find an outline of a Grade 8 Algebra I course. Keep in mind, the course content is rigorous and instruction will occur at an increased pace. Therefore, it is essential that students and parents are aware of the increased requirements of this course. NOTE: The standards in the blocks that are shaded blue are Grade 8 Standards The standards in the blocks that are shaded pink are Algebra I Standards Unit 1: Relationships between Quantities and Reasoning with Equations Clusters with Instructional Notes Reason quantitatively and use units to solve problems. Foundations for work with expressions, equations, and functions. Interpret the structure of expressions. Limit to linear expressions and to exponential expressions with integer exponents. Create equations that describe numbers or relationships. Limit #11 and #12 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit #13 to linear equations and inequalities. Limit #14 to formulas which are linear in the variables of interest. Understand solving equations as a process of reasoning and explain the reasoning. Student should focus on and master #15 for linear equations and be able to extend and apply their reasoning to other types of equations in future units and courses. Students will solve exponential equations in Algebra II. Solve equations and inequalities in one variable. AI.4. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.[n-q1] AI.5. Define appropriate quantities for the purpose of descriptive modeling.[n-q2] AI.6. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities [N-Q3] AI.7. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] a. Interpret parts of an expression, such as terms, factors, and coefficients. [A-SSE1a] b. Interpret complicated expressions by viewing one or more of their parts as a single entity. (Linear and exponential) [A-SSE1b] AI.11. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. [A-CED1] (Linear, quadratic, and exponential (integer inputs only)) AI.12. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] (Linear, quadratic, and exponential (integer inputs only)) AI.13. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. [A-CED3] AI.14. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. [A-CED4] AI.15. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. [A-REI1] AI.16. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3] Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5 x = 125.

9 Clusters with Instructional Notes Extend the properties of exponents to rational exponents. In implementing the standards in curriculum, these standards should occur before discussing exponential models with continuous domains. Analyze and solve linear equations and pairs of simultaneous linear equations. While this content is likely subsumed by standards AI.16, AI.18 and AI.19, it could be used for scaffolding instruction to the more sophisticated content found there. Solve systems of equations. (Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution). Represent and solve equations and inequalities graphically. For AI.21, focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. For AI.22, focus on cases where f(x) and g(x) are linear or exponential. Define, evaluate, and compare functions. While this content is likely subsumed by standards AI.24, AI.25, AI.26 and AI.30a, it could be used for scaffolding instruction to the more sophisticated content found there. Understand the concept of a function and use function notation. Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Constrain examples to linear functions and exponential functions having integral domains. In standard AI.26, draw connection to standard AI.34, which requires students to write arithmetic and geometric sequence. Unit 2: Linear and Exponential Functions AI.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. [N-RN1] AI.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. [N-RN2] 8.10 Analyze and solve pairs of simultaneous linear equations. [8-EE8] a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs because points of intersection satisfy both equations simultaneously. [8-EE8a] b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. [8-EE8b] c. Solve real-world and mathematical problems leading to two linear equations in two variables. [8-EE8c] AI.18. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. [A-REI5] (Linear-linear and linear-quadratic) AI.19. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [A-REI6] (Linear-linear and linear-quadratic) AI.21. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). [A-REI10] (Linear and exponential) AI.22. Explain why the x-coordinates of the points where the graphs of the equations Y= f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* [A-REI11] AI.23. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. [A-REI12] (Linear and exponential) 8.11 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) [8-F1] 8.12 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [8-F2] 8.13 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. [8-F3] AI.24. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). [F-IF1] (Focus on linear and exponential and on arithmetic and geometric sequences) AI.25. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [F-IF2] (Focus on linear and exponential and on arithmetic and geometric sequences) AI.26. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. [F-IF3]

11 Interpret expressions for functions in terms of the situation they model. sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). [F-LE2] AI.39. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. [F-LE3] AI.40. Interpret the parameters in a linear or exponential function in terms of a context. [F- LE5] (Linear and exponential of form f(x) = b x + k) Limit exponential functions to those of the form f(x) = b x + k. Clusters with Instructional Notes Summarize, represent, and interpret data on a single count or measurement variable. In grades 6-7, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution, or the existence of extreme data points. Investigate patterns of association in bivariate data. While this content is likely subsumed by standards AI.45-AI.48, it could be used for scaffolding instruction to the more sophisticated content found there. Summarize, represent, and interpret data on two categorical and quantitative variables. Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals. AI.45b should be focused on linear models, but may be used to preface quadratic functions in Unit 5 of this course. Interpret linear models. Build on students work with linear relationship and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation Unit 3: Descriptive Statistics AI.41. Represent data with plots on the real number line (dot plots, histograms, and box plots). [S-ID1] AI.42. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. [S-ID2] AI.43. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). [S-ID3] 8.25 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. [8-SP1] 8.26 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. [8-SP2] 8.27 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. [8-SP3] 8.28 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. [8-SP4] AI.44. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. [S-ID5] AI.45. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. [S-ID6] a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. [S-ID6a] b. Informally assess the fit of a function by plotting and analyzing residuals. [S-ID6b] c. Fit a linear function for a scatter plot that suggest a linear association. [S-ID6c] AI.46. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [S-ID7] AI.47. Compute (using technology) and interpret the correlation coefficient of a linear fit. [S-ID8] AI.48. Distinguish between correlation and causation. [S-ID9]

14 absolute value functions. For standard AI.36a, focus on linear but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x 2, x>0. Construct and compare linear, quadratic, and exponential models and solve problems. AI.36. Find inverse functions. [F-BF4] a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. [F-BF4a] AI.39. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. [F-LE3] Compare linear and exponential growth to quadratic growth.

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