Standard 1 Number and Computation
The student uses numerical and computational concepts and procedures in a variety of situations.
Benchmark 1
Number Sense - The student demonstrates number sense for real numbers and algebraic expressions in a variety of situations.
Indicator 1
(K) The student knows, explains, and uses equivalent representations for real numbers and algebraic expressions including integers, fractions, decimals, percents, ratios; rational number bases with integer exponents; rational numbers written in scientific notation; absolute value, time, and money (2.4.K1a) $, e.g., -4/2 = (-2); a^(-2) b^(3) = b^3/a^2.
Benchmark 2
Number Systems and Their Properties - The student demonstrates an understanding of the real number system; recognizes, applies, and explains their properties, and extends these properties to algebraic expressions.
Indicator 1
(K) The student explains and illustrates the relationship between the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using mathematical models (2.4.K1a), e.g., number lines or Venn diagrams.
Indicator 2
(K) The student identifies all the subsets of the real number system [natural counting) numbers, whole numbers, integers, rational numbers, irrational numbers] to which a given number belongs (2.4.K1m).
Indicator 3
(K) The student names, uses, and describes these properties with the real number system and demonstrates their meaning including the use of concrete objects (2.4.K1a)$:
a. commutative (a + b = b + a and ab = ba), associative [a = (b + c) = (a + b) + c and a(bc) = (ab)c], distributive [a (b + c) = ab + ac], and substitution properties (if a = 2, then 3a = 3 x 2 = 6);
b. identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a _ 1 = a, additive inverse: +5 + -5 = 0, multiplicative inverse: 8 x 1/8 = 1);
c. symmetric property of equality (if a = b, then b = a);
d. addition and multiplication properties of equality (if a = b, then a + c = b + c and if a = b, then ac = bc) and inequalities (if a > b, then a + c > b + c and if a > b, and c > 0 then ac > bc);
e. zero product property (if ab = 0, then a = 0 and/or b = 0).
Indicator 4
(K) The student uses and describes these properties with the real number system (2.4.K1a)$:
a. transitive property (if a = b and b = c, then a = c),
b. reflexive property (a = a).
Indicator 1
(A) The student generates and/or solves real-world problems with real numbers using the concepts of these properties to explain reasoning (2.4.A1a)$:
a. commutative, associative, distributive, and substitution properties, e.g., The chorus is sponsoring a trip to an amusement park. They need to purchase 15 adult tickets at $6 each and 15 student tickets at $4 each. How much money will the chorus need for tickets? The expression that models the problem is (15)(6) + (15)(4). To simplify the expression, the distributive property can be used - (15) 6) + (15)(4) = (15)(6 + 4) = (15)(10) = 150. Therefore, the chorus needs $150 to purchase the tickets.
b. identity and inverse properties, e.g., the purchase price (P) of a series EE Savings Bond is found by the formula ½ F = P where F is the face value of the bond. Use the formula to find the face value of a savings bond purchased for
$500.
½ F = P
½ F = 500
2 _1/2 F = 2(500)
1F = 1,000 or
F = 1,000c. symmetric property of equality; e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15.
d. addition and multiplication properties of equality, e.g., the total price for the purchase of three shirts in $62.54 including tax. If the tax is 3.89, what is the cost of one shirt?
T = 3s + t
$62.54 = 3s + $3.89
$62.54 - $3.89 = 3s
$58.65 = 3s
$19.55 = se. zero product property, e.g., Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning.
Indicator 2
(A) The student analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals or irrational numbers and their rational approximations in solving a given real-world problem, e.g., a store sells CDs for $12.99 each. Knowing that the sales tax is 7%, Marie estimates the cost of a CD plus tax to be $14.30. She selects nine CDs. The clerk tells Marie her bill is $157.18. How can Marie explain to the clerk she has been overcharged?
Benchmark 3
Estimation - The student uses computational estimation with real numbers in a variety of situations.
Indicator 1
(K) The student estimates real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) $.
Indicator 2
(K) The student uses various estimation strategies to estimate real number quantities and algebraic expressions $.
Indicator 3
(K) The student knows and explains why a decimal representation of an irrational number is an approximate value.
Indicator 4
(K) The student knows and explains between which two consecutive integers an irrational number lies.
Indicator 1
(A) The student adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference) $, e.g., estimate how long it takes to walk from here to there; time how long it takes to take five steps and adjust your estimate.
Indicator 2
(A) The student estimates to check whether or not the result of a real world problem using real numbers and/or algebraic expressions is reasonable and makes predictions based on the information $.
Indicator 3
(A) The student determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational strategies including mental math, paper and pencil, concrete objects, and/or appropriate technology $.
Indicator 4
(A) The student explains the impact of estimation on the result of a real world problem (underestimate, overestimate, range of estimates) $, e.g., If the weight of 25 pieces of paper was measured as 530.6 grams, what would the weight of 2000 pieces of paper equal to the nearest gram? If the student were to estimate the weight of one piece of paper as about 20 grams and then multiply this by 2,000 rather than multiply the weight of 25 pieces of paper by 80; the answer would differ by about 2,400 grams. In general, multiplying or dividing by a rounded number will cause greater discrepancies than rounding after multiplying or dividing.
Benchmark 4
Computation - The student models, performs, and explains computation with real numbers and polynomials in a variety of situations.
Indicator 1
(K) The student computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a) $.
Indicator 2
(K) The student performs and explains these computational procedures:
a. addition, subtraction, multiplication, and division using the order of operations $, N;
b. multiplication or division to find $:
i. a percent of a number, e.g., What is 0.5% of 10?;
ii. percent of increase and decrease, e.g., ex;
iii. percent one number is of another number, e.g., 89 is what percent of 82?;
iv. a number when a percent of the number is given, e.g., 80 is 32% of what number?;
c. manipulation of variable quantities within an equation or inequality (2.4.K1g), 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;
d. simplification of radical expressions (without rationalizing denominators) including square roots of perfect square monomials and cube roots of perfect cubic monomials;
e. simplification or evaluation of real numbers and algebraic monomial expressions raised to a whole number power and algebraic binomial expressions squared or cubed;
f. simplification of products and quotients of real number and algebraic monomial expressions using the properties of exponents;
g. matrix addition, e.g., a matrix could be used to compute (with one operation) a building's expenses (data) monthly, if the matrix is created to include each of the different expenses; then at the end of the year, each type of expense for the building is totaled;
h. scalar-matrix multiplication, e.g., if a matrix is created with everyone's salary in it, and everyone gets a 10% raise in pay; to find the new salary, the matrix would be multiplied by 1.1.
Indicator 3
(K) The student finds prime factors, greatest common factor, multiples, and the least common multiple of algebraic expressions (2.4.K1d).
Indicator 1
(A) The student generates and/or solves multi-step real-world problems with real numbers and algebraic expressions using computational procedures (addition, subtraction, multiplication, division, roots, and powers excluding logarithms), and mathematical concepts with:
a. applications from business, chemistry, and physics that involve addition, subtraction, multiplication, division, squares, and square roots when the formulae are given as part of the problem and variables are defined $, e.g., Given F = ma, where F = force in newtons, m = mass in kilograms, a = acceleration in meters per second squared. Find the acceleration if a force of 20 newtons is applied to a mass of 3 kilograms.;
b. volume and surface area given the measurement formulas;
c. probabilities, e.g., If the probability of getting a defective light bulb is 2%, and you buy 150 light bulbs, how many would you expect to be defective?;
d. application of percents $, e.g., compound interest given the formula;
e. simple exponential growth and decay (excluding logarithms) and economics $ e.g., A population of cells doubles every 20 years. If there are 20 cells to start with, how long will it take for there to be more than 150 cells? or If the radiation level is now 400 and it decays by ½ or its half-life is 8 hours, how long will it take for the radiation level to be below an acceptable level of 5? or If $1000 is placed in a savings account with a 6% annual interest rate and is compounded semiannually, how much money will be in the account at the end of 2 years?
The student uses algebraic concepts and procedures in a variety of situations.
Benchmark 1
Patterns - The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.
Indicator 1
(K) The student 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:
a. arithmetic and geometric sequences using real numbers and/or exponents (2.4.K1a); e.g., radioactive half-lives;
b. patterns using geometric figures (2.4.K1h);
c. 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);
d. special patterns (2.4.K1a),e.g., Pascal's triangle and the Fibonacci sequence.
Indicator 2
(K) The student generates and explains a pattern (2.4.K1f) $.
Indicator 3
(K) The student classify sequences as arithmetic, geometric, or neither.
Indicator 4
(K) The student defines (2.4.K1a)$.
a. a recursive or explicit formula for arithmetic sequences and finds any particular term,
b. a recursive or explicit formula for geometric sequences and finds any particular term.
Indicator 1
(A) The student recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written](2.4.A1i)$.
Indicator 2
(A) The student solves real-world problems with arithmetic or geometric sequences by using the explicit equation of the sequence (2.4.K1c)$,e.g., radioactive half-life, population growth, depreciation; an example of 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? and an example of 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 $1000?
Benchmark 2
Functions - The student analyzes functions in a variety of situations.
Indicator 1
(K) The student evaluates and analyzes functions using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1a,d-f).
Indicator 2
(K) The student matches equations and graphs of constant and linear functions and quadratic functions limited to y = ax^2 + c (2.4.K1d,f).
Indicator 3
(K) The student determines whether a graph, list of ordered pairs, table of values or rule represents a function (2.4.K1e-f).
Indicator 4
(K) The student determines x- and y-intercepts and maximum and minimum values of the portion of the graph that is shown on a coordinate plane (2.4.K1f).
Indicator 5
(K) The student identifies domain and range of:
a. relationships given the graph or table (2.4.K1e-f),
b. linear, constant, and quadratic functions given the equation(s) (2.4.K1d).
Indicator 6
(K) The student recognizes how changes in the constant and/or slope within a linear function changes the appearance of a graph (2.4.K1f)$.
Indicator 7
(K) The student uses function notation.
Indicator 8
(K) The student evaluates function(s) given a specific domain $.
Indicator 9
(K) The student describes the difference between independent and dependent variables and identifies independent and dependent variables $.
Indicator 1
(A) The student translates between the numerical, graphical, and symbolic representations of functions (2.4.A1c-e)$.
Indicator 2
(A) The student 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?
Indicator 3
(A) The student analyzes (2.4.A1c-e):
a. the effects of parameter changes (scale changes or restricted domains) on the appearance of a function's graph,
b. how changes in the constants and/or slope within a linear function affects the appearance of a graph,
c. how changes in the constants and/or coefficients within a quadratic function in the form of y = ax^2 + c affects the appearance of a graph.
Benchmark 3
Models - 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.
Indicator 1
(K) The student knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number lines, coordinate grids, hundred charts, measurement tools, multiplication arrays, or division sets) 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) $;
b. factor trees to model least common multiple, greatest common factor, and prime factorization; 1.4.K3);
c. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns (2.1.K1c);
d. equations and inequalities to model numerical and geometric relationships (1.4.K2c, 2.2.K3, 2.3.K1-2, 3.2.K7) $;
e. function tables to model numerical and algebraic relationships (2.1.K1c, 2.2 K2, 2.3.K1, 2.3.K3, 2.3.K5) $;
f. 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) $
g. constructions to model geometric theorems and properties; (3.1.K2, 3.1.K6);
h. 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 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);
i. scale drawings to model large and small real-world objects;
j. Pascal's Triangle to model binomial expansion and probability;
k. 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);
l. 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) $
m. Venn diagrams to sort data and to show relationships (1.2.K2).
Indicator 1
(A) The student recognizes that various mathematical models can be used to represent the same problem situation.
Mathematical models include:
a. process models (concrete objects, pictures, diagrams, flowcharts, number lines, coordinate grids, hundred charts, measurement tools, multiplication arrays or division sets) 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) $;
b. algebraic expressions to model relationships between two successive numbers in a sequence or other numerical patterns;
c. equations and inequalities to model numerical and geometric relationships (2.1.A2, 2.2.A1-3, 2.3.A1) $;
d. function tables to model numerical and algebraic relationships (2.3.A1, 2.3.A3, 3.4.A2) $;
e. 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) $;
f. 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 and isometric views of three dimensional figures (3.3 A1, 4.2.A1c);
g. scale drawings to model large and small real-world objects (3.3.A3, 3.4.A3);
h. geometric models (spinners, targets, or number cubes) and process models coins, pictures, or diagrams) to model probability (1.4.A1c, 4.2.A1, 4.2.A3);
i. 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) $;
j. Venn diagrams to sort data and show relationships.
Indicator 2
(A) The student uses the mathematical modeling process to analyze and make inferences about real-world situations $.
The student uses geometric concepts and procedures in a variety of situations.
Benchmark 1
Geometric Figures and Their Properties - The student recognizes geometric figures and compares properties and concepts of geometric figures, and justifies the properties of geometric figures in a variety of situations.
Indicator 1
(K) The student recognizes and compares properties of two-and three dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology (2.4.K1h).
Indicator 2
(K) The student discusses properties of regular polygons related to (2.4.K1g-h):
a. angle measures,
b. diagonals.
Indicator 3
(K) The student recognizes and describes the symmetries (point, line, plane) that exist in three-dimensional figures (2.4.K1h).
Indicator 4
(K) The student recognizes that similar figures have congruent angles, and their corresponding sides are proportional (2.4.K1h).
Indicator 5
(K) The student uses the Pythagorean Theorem to (2.4.K1h):
a. determine if a triangle is a right triangle,
b. find a missing side of a right triangle.
Indicator 6
(K) The student recognizes and describes (2.4.K1g-h):
a. congruence of triangles using: Side-Side-Side (SSS), Angle-Side Angle (ASA) Side-Angle-Side (SAS), and Angle-Angle-Side (AAS);
b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°.
Indicator 7
(K) The student recognizes, describes, and compares the relationships of the angles formed when parallel lines are cut by a transversal (2.4.K1h).
Indicator 8
(K) The student recognizes and identifies parts of a circle: arcs, chords, sectors of circles, secant and tangent lines, central and inscribed angles (2.4.K1h).
Indicator 1
(A) The student solves real-world problems by (2.4.A1a):
a. using the properties of corresponding parts of similar and congruent figures, e.g., scale drawings, map reading, or proportions;
b. applying the Pythagorean Theorem, e.g., When checking for square corners on concrete forms for a foundation, determine if a right angle is formed by using the Pythagorean Theorem;
c. using properties of parallel lines, e.g., street intersections.
Indicator 2
(A) The student uses deductive reasoning to justify the relationships between the sides of 30°-60°-90° and 45°-45°-90° triangles using the ratios of sides of similar triangles (2.4.A1a).
Indicator 3
(A) The student understands the concepts of and develops a formal or informal proof through understanding of the difference between a statement verified by proof (theorem) and a statement supported by examples (2.4.A1a).
Benchmark 2
Measurement and Estimation - The student estimates, measures and uses geometric formulas in a variety of situations.
Indicator 1
(K) The student determines and uses real number approximations (estimations) for length, width, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement using standard and nonstandard units of measure (2.4.K1a) $.
Indicator 2
(K) The student selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, distance, area, surface area, mass, midpoint, and angle measurements (2.4.K1a) $.
Indicator 3
(K) The student approximates conversions between customary and metric systems given the conversion unit or formula (2.4.K1a).
Indicator 4
(K) The student states, recognizes, and applies formulas for (2.4.K1h) $:
a. perimeter and area of squares, rectangle, and triangles;
b. circumference and area of circles;
c. volume of rectangular solids.
Indicator 5
(K) The student uses given measurement formulas to find perimeter, area, volume, and surface area of two- and three-dimensional figures (regular and irregular) (2.4.K1h).
Indicator 6
(K) The student recognizes and applies properties of corresponding parts of similar and congruent figures to find measurements of missing sides (2.4.K1a).
Indicator 7
(K) The student knows, explains, and uses ratios and proportions to describe rates of change (2.4.K1d) $, e.g., miles per gallon, meters per second, calories per ounce, or rise over run.
Indicator 1
A) The student solves real-world problems by:
a. converting within the customary and the metric systems, e.g., Marti and Ginger are making a huge batch of cookies and so they are multiplying their favorite recipe quite a few times. They find that they need 45 tablespoons of liquid. To the nearest ¼ of a cup, how many cups would be needed?
b. finding the perimeter and the area of circles, squares, rectangles, triangles, parallelograms, and trapezoids, e.g., a track is made up of a rectangle with dimensions 100 meters by 50 meters with semicircles at each end (having a diameter of 50 meters). What is the distance of one lap around the inside lane of the track?
c. finding the volume and the surface area of rectangular prisms and cylinders, e.g., if a car engine has 6 cylinders and each cylinder has a height of 8.4 cm and a diameter of 8.8 cm, then what is the total volume of the cylinders?
d. using the Pythagorean theorem, e.g., a baseball diamond is a square with 90 feet between each base. What is the approximate distance from home plate to second base?
e. using rates of change, e.g., the equation w = -52 + 1.6t can be used to approximate the wind chill temperatures for a wind speed of 40 mph. Find the wind chill temperature (w) when the actual temperature (t) is 32 degrees. What part of the equation represents the rate of change?
Indicator 2
(A) The student estimates to check whether or not measurements or calculations for length, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) $.
Indicator 3
(A) The student uses indirect measurements to measure inaccessible objects (2.4.A1a), , e.g., You are standing next to the railroad tracks and a train passes. The number of cars in the train can be determined if you know how long it takes for one car to pass and the length of time the whole train takes to pass you.
Benchmark 3
Transformational Geometry - The student recognizes and applies transformations on two- and three-dimensional figures in a variety of situations.
Indicator 1
(K) The student describes and performs single and multiple transformations [refection, rotation, translation, reduction (contraction/shrinking), enlargement magnification/growing)] on two- and three-dimensional figures (2.4.K1a).
Indicator 2
(K) The student recognizes a three-dimensional figure created by rotating a simple two-dimensional figure around a fixed line ((2.4.K1a),e.g., a rectangle rotated about one of its edges generates a cylinder; an isosceles triangle rotated about a fixed line that runs from the vertex to the midpoint of its base generates a cone.
Indicator 3
(K) The student generates a two-dimensional representation of a three dimensional figure (2.4.K1a).
Indicator 4
(K) The student determines where and how an object or a shape can be tessellated using single or multiple transformations and creates a tessellation (2.4.K1a).
Indicator 1
(A) The student analyzes the impact of transformations on the perimeter and area of circles, rectangles, and triangles and volume of rectangular prisms and cylinders (2.4.A1f),e.g., reducing by a factor of ½ multiplies an area by a factor of ¼ and multiplies the volume by a factor of 1/8, whereas, rotating a geometric figure does not change perimeter or area.
Indicator 2
(A) The student describes and draws a simple three-dimensional shape after undergoing one specified transformation without using concrete objects to perform the transformation (2.4.A1a).
Indicator 3
(A) The student uses a variety of scales to view and analyze two- and three-dimensional figures (2.4.A1g).
Indicator 4
(A) The student analyzes and explains transformations using such things as sketches and coordinate systems (2.4.A1a).
Benchmark 4
Geometry from an Algebraic Perspective - The student uses an algebraic perspective to analyze the geometry of two- and three-dimensional figures in a variety of situations.
Indicator 1
(K) The student recognizes and examines two- and three-dimensional figures and their attributes including the graphs of functions on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.K1f).
Indicator 2
(K) The student determines if a given point lies on the graph of a given line or parabola without graphing and justifies the answer (2.4.K1f).
Indicator 3
(K) The student calculates the slope of a line from a list of ordered pairs on the line and explains how the graph of the line is related to its slope (2.4.K1f).
Indicator 4
(K) The student finds and explains the relationship between the slopes of parallel and perpendicular lines lines (2.4.K1f), e.g., 2x + 3y = 12 name a linear function. What is the slope of the line that is formed by this equation? Write and equation of a perpendicular to 2x + 3y = 12. Explain how the slopes of all three of these lines relate to each other.
Indicator 5
(K) The student uses the Pythagorean Theorem to find distance (may use the distance formula) (2.4.K1f).
Indicator 6
(K) The student recognizes the equation of a line and transforms the equation into slope-intercept form in order to identify the slope and y intercept and uses this information to graph the line (2.4.K1f).
Indicator 7
(K) The student recognizes the equation y = ax^2 + c as a parabola; represents and identifies characteristics of the parabola including opens upward or opens downward, steepness (wide/narrow), the vertex, maximum and minimum values, and line of symmetry; and sketches the graph of the parabola (2.4.K1f).
Indicator 8
(K) The student explains the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs (2.4.K1f),e.g., for equations, the lines intersect in either one point, no points, or infinite points; and for inequalities, all points in double-shaded areas are solutions for both inequalities.
Indicator 1
(A) The student represents, generates, and/or solves real-world problems that involve distance and two-dimensional geometric figures including parabolas in the form ax^2 + c (2.4.A1e),e.g., compare the heights of 2 different objects whose paths are represented h1( t) = 3 t ² + 1 and h2( t) = ½ t ² + 4 (where h represents the height in feet and t represents elapsed time in seconds) during the interval from 0-5 seconds.
Indicator 2
(A) The student translates between the written, numeric, algebraic, and geometric representations of a real-world problem (2.4.A1a-e) $, e.g., given a situation (ex), write a function rule, make a T-table of the algebraic relationship, graph the order pairs.
Indicator 3
(A) The student recognizes and explains the effects of scale changes on the appearance of the graph of an equation involving a line or parabola (2.4.A1g).
Indicator 4
(A) The student analyzes how changes in the constants and/or leading coefficients within the equation of a line or parabola affects the appearance of the graph of the equation (2.4.A1e).
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