Standard
Benchmark
Indicator | Description | Lesson Plans | Thinkfinity | Resources |
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2
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The student uses algebraic concepts in a variety of situations.
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2.1
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The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.
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2.1.A1
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recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1i) ($).
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2.1.K1
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identifies, states, and continues the following patterns using various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written
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2.1.K1A
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arithmetic and geometric sequences using real numbers and/or exponents (2.4.K1a); e.g., radioactive half-lives;
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2.1.K1B
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patterns using geometric figures (2.4.K1h);
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2.1.K1C
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algebraic patterns including consecutive number patterns or equations of functions, e.g., n, n + 1, n + 2, ... or f(n) = 2n – 1 (2.4.K1c,e);
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2.1.K1D
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special patterns (2.4.K1a), e.g., Pascal’s triangle and the Fibonacci sequence.
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2.1.K2
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generates and explains a pattern (2.4.K1f).
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2.1.A2
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solves real-world problems with arithmetic or geometric sequences by using the explicit equation of the sequence (2.4.K1c) ($), e.g., an arithmetic sequence: A brick wall is 3 feet high and the owners want to build it higher. If the builders can lay 2 feet every hour, how long will it take to raise it to a height of 20 feet? or a geometric sequence: A savings program can double your money every 12 years. If you place $100 in the program, how many years will it take to have over $1,000?
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2.1.K3
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classify sequences as arithmetic, geometric, or neither.
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2.1.K4
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defines (2.4.K1a):
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2.1.K4A
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a recursive or explicit formula for arithmetic sequences and finds any particular term,
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2.1.K4B
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a recursive or explicit formula for geometric sequences and finds any particular term.
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2.2
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The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.
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2.2.A1
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represents real-world problems using variables, symbols, expressions, equations, inequalities, and simple systems of linear equations (2.4.A1c-e) ($).
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2.2.K1
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knows and explains the use of variables as parameters for a specific variable situation (2.4.K1f), e.g., the m and b in y = mx + b or the h, k, and r in (x – h)2 + (y – k)2 = r2.
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2.2.K2
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manipulates variable quantities within an equation or inequality (2.4.K1e), e.g., 5x – 3y = 20 could be written as 5x – 20 = 3y or 5x(2x + 3) = 8 could be written as 8/(5x) = 2x + 3.
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2.2.A2
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represents and/or solves real-world problems with (2.4.A1c) ($):
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2.2.A2A
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linear equations and inequalities both analytically and graphically, e.g., tickets for a school play are $5 for adults and $3 for students. You need to sell at least $65 in tickets. Give an inequality and a graph that represents this situation and three possible solutions.
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2.2.A2B
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quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring), e.g., a fence is to be built onto an existing fence. The three sides will be built with 2,000 meters of fencing. To maximize the rectangular area, what should be the dimensions of the fence?
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2.2.A2C
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systems of linear equations with two unknowns, e.g., when comparing two cellular telephone plans, Plan A costs $10 per month and $.10 per minute and Plan B costs $12 per month and $.07 per minute. The problem is represented by Plan A = .10x + 10 and Plan B = .07x + 12 where x is the number of minutes.
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2.2.A2D
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radical equations with no more than one inverse operation around the radical expression, e.g., a square rug with an area of 200 square feet is 4 feet shorter than a room. What is the length of the room?
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2.2.A2E
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a rational equation where the solution can be simplified as a linear equation with a nonzero denominator, e.g., John is 2 feet taller than Fred. John’s shadow is 6 feet in length and Fred’s shadow is 4 feet in length. How tall is Fred?
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2.2.A3
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explains the mathematical reasoning that was used to solve a real-world problem using equations and inequalities and analyzes the advantages and disadvantages of various strategies that may have been used to solve the problem (2.4.A1c).
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2.2.K3
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solves (2.4.K1d) ($):
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2.2.K3A
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linear equations and inequalities both analytically and graphically;
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2.2.K3B
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quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring);
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2.2.K3C
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systems of linear equations with two unknowns using integer coefficients and constants;
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2.2.K3D
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radical equations with no more than one inverse operation around the radical expression;
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2.2.K3E
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equations where the solution to a rational equation can be simplified as a linear equation with a nonzero denominator.
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2.2.K3F
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equations and inequalities with absolute value quantities containing one variable with a special emphasis on using a number line and the concept of absolute value.
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2.2.K3G
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exponential equations with the same base without the aid of a calculator or computer, e.g., 3x + 2 = 35
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2.3
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The student analyzes functions in a variety of situations.
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2.3.A1
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translates between the numerical, graphical, and symbolic representations of functions (2.4.A1c-e) ($).
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2.3.K1
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evaluates and analyzes functions using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1a, d-f).
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2.3.K2
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matches equations and graphs of constant and linear functions and quadratic functions limited to y = ax2 + c (2.4.K1d, f).
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2.3.A2
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interprets the meaning of the x- and y- intercepts, slope, and/or points on and off the line on a graph in the context of a real-world situation (2.4.A1e) ($), e.g., the graph below represents a tank full of water being emptied. What does the y-intercept represent? What does the x-intercept represent? What is the rate at which it is emptying? What does the point (2, 25) represent in this situation? What does the point (2,30) represent in this situation?
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2.3.A3
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analyzes (2.4.A1c-e):
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2.3.A3A
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the effects of parameter changes (scale changes or restricted domains) on the appearance of a function’s graph,
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2.3.A3B
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how changes in the constants and/or slope within a linear function affects the appearance of a graph,
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2.3.A3C
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how changes in the constants and/or coefficients within a quadratic function in the form of y = ax2 + c affects the appearance of a graph.
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2.3.K3
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determines whether a graph, list of ordered pairs, table of values, or rule represents a function (2.4.K1e-f).
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2.3.K4
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determines x- and y-intercepts and maximum and minimum values of the portion of the graph that is shown on a coordinate plane (2.4.K1f).
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2.3.K5
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identifies domain and range of:
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2.3.K5A
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relationships given the graph or table (2.4.K1e-f),
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2.3.K5B
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linear, constant, and quadratic functions given the equation(s) (2.4.K1d).
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2.3.K6
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recognizes how changes in the constant and/or slope within a linear function changes the appearance of a graph (2.4.K1f) ($).
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2.3.K7
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uses function notation.
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2.3.K8
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evaluates function(s) given a specific domain ($).
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2.3.K8
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describes the difference between independent and dependent variables and identifies independent and dependent variables ($).
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2.4
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The student develops and uses mathematical models to represent and justify mathematical relationships found in a variety of situations involving tenth grade knowledge and skills.
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2.4.A1
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recognizes that various mathematical models can be used to represent the same problem situation. Mathematical models include:
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2.4.A1A
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process models (concrete objects, pictures, diagrams, flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.K1, 1.2.A1-2, 1.3.A1-4, 1.4.A1a, 1.4A1d-e, 3.1.A1-3, 3.2.A1-3, 3.3.A2, 3.3.A4, 3.4.A2, 4.2.A1a-b) ($);
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2.4.A1B
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algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns;
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2.4.A1C
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equations and inequalities to model numerical and geometric relationships (2.1.A2, 2.2.A1-3, 2.3.A1) ($);
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2.4.A1D
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function tables to model numerical and algebraic relationships (2.3.A1, 2.3.A3, 3.4.A2) ($);
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2.4.A1E
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coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.2.A1, 2.3.A1-3, 3.4.A1-2, 3.4.A4) ($);
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2.4.A1F
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two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area, volume, and surface area, properties of two- and three-dimensional figures and isometric views of three-dimensional figures (3.3.A1, 4.2.A1c);
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2.4.A1G
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scale drawings to model large and small real-world objects (3.3.A3, 3.4.A3);
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2.4.A1H
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geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (1.4.A1c, 4.2.A1, 4.2.A3);
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2.4.A1I
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frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, histograms, and matrices to describe, interpret, and analyze data (2.1.A1, 4.1.A1, 4.1.A3-4, 4.1.A6, 4.2.A1) ($);
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2.4.A1J
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Venn diagrams to sort data and show relationships.
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2.4.K1
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knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships. Mathematical models include:
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2.4.K1A
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process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-3, 1.2.K1, 1.2.K3-4, 1.3.K1-4, 1.4.K1, 1.4.K2a-b, 2.1.K1a, 2.1.K1d, 2.1.K2, 2.2.K4, 2.3.K1, 3.2.K1-3, 3.2.K6, 3.3.K1-4, 4.2.K3-4) ($);
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2.4.K1B
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factor trees to model least common multiple, greatest common factor, and prime factorization (1.4.K3);
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2.4.K1C
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algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns (2.1.K1c);
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2.4.K1D
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equations and inequalities to model numerical and geometric relationships (1.4.K2c, 2.2.K3, 2.3.K1-2, 3.2.K7) ($);
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2.4.K1E
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function tables to model numerical and algebraic relationships (2.1.K1c, 2.2.K2, 2.3.K1, 2.3.K3, 2.3.K5) ($);
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2.4.K1F
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coordinate planes to model relationships between ordered pairs and equations and inequalities and linear and quadratic functions (2.2.K1, 2.3.K1-6, 3.4.K1-8) ($) ;
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2.4.K1G
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constructions to model geometric theorems and properties (3.1.K2, 3.1.K6);
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2.4.K1H
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two- and three-dimensional geometric models (geoboards, dot paper, coordinate plane, nets, or solids) and real-world objects to model perimeter, area, volume, and surface area, properties of two- and three-dimensional figures, and isometric views of three-dimensional figures (2.1.K1b, 3.1.K1-8, 3.2.K1, 3.2.K4-5, 3.3.K1-4);
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2.4.K1I
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scale drawings to model large and small real-world objects;
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2.4.K1J
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Pascal’s Triangle to model binomial expansion and probability;
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2.4.K1K
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geometric models (spinners, targets, or number cubes), process models (concrete objects, pictures, diagrams, or coins), and tree diagrams to model probability (4.1.K1-3);
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2.4.K1L
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frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, histograms, and matrices to organize and display data (4.2.K1, 4.2.K5-6) ($);
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2.4.K1M
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Venn diagrams to sort data and show relationships (1.2.K2).
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2.4.A2
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uses the mathematical modeling process to analyze and make inferences about real-world situations ($).
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