| 2.8. Algebra and Functions |
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| 2.8.3. GRADE 3 | 2.8.5. GRADE 5 | 2.8.8. GRADE 8 | 2.8.11. GRADE 11 |
| Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to: |
| A. | Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns. | A. | Recognize, reproduce, extend, create and describe patterns, sequences and relationships verbally, numerically, symbolically and graphically, using a variety of materials. | A. | Apply simple algebraic patterns to basic number theory and to spatial relations. | A. | Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.
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| B. | Use concrete objects and trial and error to solve number sentences and check if solutions are sensible and accurate. | B. | Connect patterns to geometric relations and basic number skills. | B. | Discover, describe and generalize patterns, including linear, exponential and simple quadratic relationships. | B. | Give examples of patterns that occur in data from other disciplines. |
| C. | Substitute a missing addend in a number sentence. | C. | Form rules based on patterns (e.g., an equation that relates pairs in a sequence). | C. | Create and interpret expressions, equations or inequalities that model problem situations. | C. | Use patterns, sequences and series to solve routine and nonroutine problems. |
| D. | Create a story to match a given combination of symbols and numbers. | D. | Use concrete objects and combinations of symbols and numbers to create expressions that model mathematical situations. | D. | Use concrete objects to model algebraic concepts. | D. | Formulate expressions, equations, inequalities, systems of equations, systems of inequalities and matrices to model routine and nonroutine problem situations. |
| E. | Use concrete objects and symbols to model the concepts of variables, expressions, equations and inequalities. | E. | Explain the use of combinations of symbols and numbers in expressions, equations and inequalities. | E. | Select and use a strategy to solve an equation or inequality, explain the solution and check the solution for accuracy. | E. | Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas). |
| F. | Explain the meaning of solutions and symbols. | F. | Describe a realistic situation using information given in equations, inequalities, tables or graphs. | F. | Solve and graph equations and inequalities using scientific and graphing calculators and computer spreadsheets. | F. | Identify whether systems of equations and inequalities are consistent or inconsistent. |
| G. | Use a table or a chart to display information. | G. | Select and use appropriate strategies, including concrete materials, to solve number sentences and explain the method of solution. | G. | Represent relationships with tables or graphs in the coordinate plane and verbal or symbolic rules. | G. | Analyze and explain systems of equations, systems of inequalities and matrices. |
| H. | Describe and interpret the data shown in tables and charts. | H. | Locate and identify points on a coordinate system. | H. | Graph a linear function from a rule or table. | H. | Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets and other software. |
| I. | Demonstrate simple function rules. | I. | Generate functions from tables of data and relate data to corresponding graphs and functions. | I. | Generate a table or graph from a function and use graphing calculators and computer spreadsheets to graph and analyze functions. | I. | Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication and scalar multiplication. |
| J. | Analyze simple functions and relationships and locate points on a simple grid. | | | J. | Show that an equality relationship between two quantities remains the same as long as the same change is made to both quantities; explain how a change in one quantity determines another quantity in a functional relationship. | J. | Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane. |
| | | | | | K. | Select, justify and apply an appropriate technique to graph a linear function in two variables, including slope-intercept, x- and y-intercepts, graphing by transformations and the use of a graphing calculator. |
| | | | | | L. | Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line. |
| | | | | | M. | Given a set of data points, write an equation for a line of best fit. |
| | | | | | N. | Solve linear, quadratic and exponential equations both symbolically and graphically. |
| | | | | | O. | Determine the domain and range of a relation, given a graph or set of ordered pairs. |
| | | | | | P. | Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically. |
| | | | | | Q. | Represent functional relationships in tables, charts and graphs. |
| | | | | | R. | Create and interpret functional models. |
| | | | | | S. | Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic). |
| | | | | | T. | Analyze and categorize functions by their characteristics. |
| 2.9. Geometry |
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| 2.9.3 GRADE 3 | 2.9.5. GRADE 5 | 2.9.8 GRADE 8 | 2.9.11 GRADE 11 |
| Pennsylvania's public school shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to: |
| A. | Name and label geometric shapes in two and three dimensions (e.g., circle/sphere, square/cube, triangle/pyramid, rectangle/prism). | A. | Give formal definitions of geometric figures. | A. | Construct figures incorporating perpendicular and parallel lines, the perpendicular bisector of a line segment and an angle bisector using computer software. | A. | Construct geometric figures using dynamic geometry tools (e.g., Geometer's Sketchpad, Cabri Geometre). |
| B. | Build geometric shapes using concrete objects (e.g., manipulatives). | B. | Classify and compare triangles and quadrilaterals according to sides or angles. | B. | Draw, label, measure and list the properties of complementary, supplementary and vertical angles. | B. | Prove two triangles or two polygons are congruent or similar using algebraic, coordinate and deductive proofs. |
| C. | Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on the geoboard and on graph paper satisfying specific criteria. | C. | Identify and measure circles, their diameters and radii. | C. | Classify familiar polygons as regular or irregular up to a decagon. | C. | Identify and prove the properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles and diagonals using deductive proofs. |
| D. | Find and describe geometric figures in real life. | D. | Describe in words how geometric shapes are constructed. | D. | Identify, name, draw and list all properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms and cylinders. | D. | Identify corresponding parts in congruent triangles to solve problems. |
| E. | Identify and draw lines of symmetry in geometric figures. | E. | Construct two- and three-dimensional shapes and figures using manipulatives, geoboards and computer software. | E. | Construct parallel lines, draw a transversal and measure and compare angles formed (e.g., alternate interior and exterior angles). | E. | Solve problems involving inscribed and circumscribed polygons. |
| F. | Identify symmetry in nature. | F. | Find familiar solids in the environment and describe them. | F. | Distinguish between similar and congruent polygons. | F. | Use the properties of angles, arcs, chords, tangents and secants to solve problems involving circles. |
| G. | Fold paper to demonstrate the reflections about a line. | G. | Create an original tessellation. | G. | Approximate the value of pi through experimentation. | G. | Solve problems using analytic geometry. |
| H. | Show relationships between and among figures using reflections. | H. | Describe the relationship between the perimeter and area of triangles, quadrilaterals and circles. | H. | Use simple geometric figures (e.g., triangles and squares) to create, through rotation, transformational figures in three dimensions. | H. | Construct a geometric figure and its image using various transformations. |
| I. | Predict how shapes can be changed by combining or dividing them. | I. | Represent and use the concepts of line, point and plane. | I. | Generate transformations using computer software. | I. | Model situations geometrically to formulate and solve problems. |
| | J. | Define the basic properties of squares, pyramids, parallelograms, quadrilaterials, trapezoids, polygons, rectangles, rhombi, circles, triangles, cubes, prisms, spheres and cylinders. | J. | Analyze geometric patterns (e.g., tessellations and sequences of shapes) and develop descriptions of the patterns. | J. | Analyze figures in terms of the kinds of symmetries they have. |
| | K. | Analyze simple transformations of geometric figures and rotations of line segments. | K. | Analyze objects to determine if they illustrate tessellations, symmetry, congruence, similarity and scale. | | |
| | L. | Identify properties of geometric figures (e.g., parallel, perpendicular, similar, congruent, symmetrical). | | | | |
| 2.11. Concepts of Calculus |
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| 2.11.3 GRADE 3 | 2.11.5. GRADE 5 | 2.11.8 GRADE 8 | 2.11.11 GRADE 11 |
| Pennsylvania's public schools shall teach, challenge and support every student to realize the student's maximum potential and to acquire the knowledge and skills to: |
| A. | Identify whole number quantities and measurements from least to most and greatest value. | A. | Make comparisons of numbers (e.g., more, less, same, least, most, greater than, less than). | A. | Analyze graphs of related quantities for minimum and maximum values and justify the findings. | A. | Determine maximum and minimum values of a function over a specified interval. |
| B. | Identify least and greatest values represented in bar graphs and pictographs. | B. | Identify least and greatest values represented in bar and circle graphs. | B. | Describe the concept of unit rate, ratio, and slope in context of rate of change. | B. | Interpret maximum and minimum values in problem situations. |
| C. | Categorize rates of change as faster and slower. | C. | Identify maximum and minimum. | C. | Continue a pattern of numbers or objects that could be extended infinitely. | C. | Graph and interpret rates of growth/decay. |
| D. | Continue a pattern of numbers or objects that could be extended infinitely. | D. | Describe the relationship between rates of change and time. | | | D. | Determine sums of finite sequences of numbers and infinite geometric series. |
| | E. | Estimate areas and volumes as the sums of areas of tiles and volumes of cubes. | | | E. | Estimate areas under curves using sequences of areas. |
| | F. | Describe the relationship between the size of the unit of measurement and the estimate of the areas and volumes. | | | | |
| Combination: | A subset of the elements in a given set, without regard to the order in which those elements are arranged. |
| Composite number: | Any positive integer exactly divisible by one or more positive integers other than itself and 1. |
| Congruent: | Having the same shape and the same size. |
| Conjecture: | A statement believed to be true but not proved. |
| Coordinate system: | A method of locating points in the plane or in space by means of numbers. A point in the plane is located by its distances from both a horizontal and a vertical line called the axes. The horizontal line is called the x-axis. The vertical line is called the y-axis. The pairs of numbers are called ordered pairs. The first number, called the x-coordinate, designates the distance along the horizontal axis. The second number, called the y-coordinate, designates the distance along the vertical axis. The point at which the two axes intersect has the coordinates (0,0) and is called the origin.
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| Correlation: | A measure of the mutual relationship between two variables. |
| Customary system: | A system of weights and measures frequently used in the United States. The basic unit of weight is the pound; the basic unit of capacity is the quart. |
| Deductive reasoning: | The process of reasoning from statements accepted as true to reach a conclusion. |
| Direct variation: | Two variables are so related that their ratio remains constant. |
| Domain: | The set of all possible values for the unknown in an open sentence. |
| Equation: | A statement of equality between two mathematical expressions (e.g., X + 5 = Y - 2). |
| Equivalent forms: | Different forms of numbers that name the same number (e.g., fraction, decimal, percent as 1/2, .5, 50%). |
| Expanded notation: | Involves writing the number in expanded form to show the value of each digit (e.g., 15,629 = 10,000 + 5,000 + 600 + 20 + 9). |
| Exponential function: | A function whose general equation is y = a × bx or y = a × bkx, where a, b and k stand for constants. |
| Exponent: | A numeral used to tell how many times a number or variable is used as a factor (e.g., a2, 2n, yx). |
| Expression: | A mathematical phrase that can include operations, numerals and variables. In algebraic terms: 2l + 3x; in numeric terms: 13.4 - 4.7. |
| Factor: | The number or variable multiplied in a multiplication expression. |
| Factorial: | The expression n! (n factorial) is the product of all the numbers from 1 to n for any positive integer n. |
| Function: | A relation in which each value of an independent variable is associated with a unique value of the dependent value. |
| Geoboard: | A board with pegs aligned in grid fashion that permits rubber bands to be wrapped around pegs to form geometric figures. |
| Graphing calculator: | A calculator that will store and draw the graphs of several functions at once. |
| Independent events: | Events such that the outcome of the first event has no effect on the probabilities of the outcome of the second event (e.g., two tosses of the same coin are independent events). |
| Inductive reasoning: | Generalizations made from particular observations in a common occurrence. |
| Inequality: | A mathematical sentence that contains a symbol, (e.g., >, <, >=, <= or <>) in which the terms on either side of the symbol are unequal (e.g., x < y, 7 > 3, n >= 4). |
| Infinite: | Has no end or goes on forever. |
| Integer: | A number that is a positive whole number, a negative whole number or zero. |
| Inverse: | A new conditional formed by negating both the antecedent and the consequent of a conditional. |
| Inverse operations: | Operations that undo each other (e.g., addition and subtraction are inverse operations; multiplication and division are inverse operations). |
| Inverse variation: | When the ratio of one variable to the reciprocal of the other is constant, one of them is said to vary inversely as the other. |
| Irrational number: | A number that cannot be written as a simple fraction. It is an infinite and nonrepeating decimal. |
| Limit: | A number to which the terms of a sequence get closer so that beyond a certain term all terms are as close as desired to that number. |
| Line of best fit: | The line that fits a set of data points with the smallest value for the sum of the squares of the errors (vertical distances) from the data points to the line; the regression line. |
| Linear function: | A function whose general equation is y = mx + b, where m and b stand for constants and m <> 0. |
| Linear measurement: | Measurement in a straight line. |
| Logarithm: | The exponent indicating the power to which a fixed number, the base, must be raised to produce a given number. For example, if nx = a, the logarithm of a, with n as the base, is x; symbolically, logna = x. If the base is 10, the log of 100 is 2. |
| Manipulatives: | Materials that allow students to explore mathematical concepts in a concrete mode. |
| Mathematical model: | A representation in the mathematical world of some phenomenon in the real world. It frequently consists of a function or relation specifying how two variables are related. |
| Matrix: | A rectangular array of numbers representing such things as the coefficients in a system of equations arranged in rows and columns. |
| Maximum: | The greatest number in a set of data. |
| Mean: | The sum of the set of numbers divided by n, the number of numbers in the set. |
| Median: | The number that lies in the middle when a set of numbers is arranged in order. If there are two middle values, the median is the mean of these values. |
| Metric system: | A system of measurement used throughout the world based on factors of 10. It includes measures of length, weight and capacity. |
| Minimum: | The least number in a set of data. |
| Missing addend: | A member of an addition number sentence in which that term is missing (e.g., 5 + __= 8). |
| Mode: | The number(s) that occurs most often in a set of numbers (e.g., in the set 1, 2, 3, 3, 5, 8; the mode is 3). |
| Multiple: | A number that is the product of a given integer and another integer (e.g., 6 and 9 are multiples of 3). |
| Normal curve: | A graphical plot of a mathematical function (frequency distribution) which is unimodal and symmetrical. |
One-to-one
correspondence: | When one and only one element of a second set is assigned to an element of a first set, all elements of the second set are assigned, and every element of the first set has an assignment, the mapping is called one-to-one (e.g., in the set Bill Clinton, George Bush, Ronald Reagan, Jimmy Carter, Hillary Clinton, Barbara Bush, Nancy Reagan and Rosalynn Carter, there is a one-to-one correspondence between the pairs.) |
| Open sentence: | A statement that contains at least one unknown. It becomes true or false when a quantity is substituted for the unknown (e.g., x + 5 = 9, y - 2 = 7). |
| Order of operations: | Rules for evaluating an expression: work first within parentheses; then calculate all powers, from left to right; then do multiplications or divisions, from left to right; then do additions and subtractions, from left to right. |
| Patterns: | Regularities in situations such as those in nature, events, shapes, designs and sets of numbers (e.g., spirals on pineapples, geometric designs in quilts, the number sequence 3, 6, 9, 12, . . ). |
| Permutation: | An arrangement of a given number of objects from a given set in which the order of the objects is significant. |
| Perpendicular lines: | Two lines that intersect to form right angles.
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| Plotting points: | Locating points by means of coordinates, or a curve by plotted points, representing an equation by means of a curve so constructed. |
| Polygon: | A union of segments connected end to end, so that each segment intersects exactly two others at its endpoints. |
| Powers: | A number expressed using an exponent. The number 53 is read five to the third power or five cubed. |
| Prime: | An integer greater than one whose only positive factors are 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17 and 19). |
| Probability: | A number from 0 to 1 that indicates how likely something is to happen. |
| Problem solving: | Finding ways to reach a goal when no routine path is apparent. |
Proof by
contradiction: | A proof in which, if s is to be proven, one reasons from not-s until a contradiction is deduced; from this it is concluded that not-s is false, which means that s is true. |
| Proportion: | An equation of the form a/b = c/d that states that the two ratios are equivalent. |
| Quadrilateral: | A four-sided polygon. |
| Quartiles: | The three values that divide an ordered set into four subsets of approximately equal size. The second quartile is the median. |
| Radian: | A unit of angular measure equal to ½ pi of a complete revolution. |
| Range (1): | The difference between the greatest number and the least number in a set of data. |
| Range (2): | The set of output values for a function. |
| Rate of change: | The limit of the ratio of an increment of the function value at the point to that of the independent variable as the increment of the variable approaches zero. |
| Ratio: | A comparison of two numbers by division. |
| Rational numbers: | Any number that can be written in the form a/b where a is any interger and b is any integer except zero. |
| Real numbers: | The set consisting of all rational numbers and all irrational numbers. |
| Reasonableness: | Quality of a solution so that it is not extreme or excessive. |
| Reciprocal: | The fractional number that results from dividing one by the number. |
| Rectangular prism: | A three-dimensional figure whose sides are all rectangles; a box. |
| Reflection: | A transformation that produces the mirror image of a geometric figure. |
| Regression: | The line that represents the least deviation from the points in a scatter plot of data. |
| Regular polygon: | A polygon in which all sides have the same measure and all angles have the same measure. |
| Relation: | A set of ordered pairs. |
| Reliability: | The extent to which a measuring procedure yields the same results on repeated trials. |
| Repeated addition: | A model for multiplication (e.g., 2 + 2 + 2 = 3 x 2). |
| Rotation: | A transformation that maps every point in the plane to its image by rotating the plane around a fixed point or line. |
| Scientific calculator: | A calculator that represents very large or very small numbers in scientific notation and with the powering, factorial, square root, negative and reciprocal keys. |
| Scientific notation: | A way of writing a number of terms of an integer power of 10 multiplied by a number greater than or equal to 1 and less than 10. |
| Sequence: | A set of ordered quantities (e.g., positive integers). |
| Series: | The indicated sum of the terms of a sequence. |
| Similarity: | Having the same shape but not necessarily the same size. |
| Simple event: | An event whose probability can be obtained from consideration of a single occurrence (e.g., the tossing of a coin is a simple event). |
| Simulation: | Modeling a real event without actually observing the event. |
| Slope: | The slope of a line is the ratio of the change in y to the corresponding change in x; the constant m in the linear function equation; rise/run. |
| Standard deviation: | The square root of the variance. |
| Stem-and-leaf plot: | A frequency distribution made by arranging data (e.g., student scores on a test were 98, 96, 85, 93, 83, 87, 85, 87, 93, 75, 77 and 83. This data are displayed in a stem-and-leaf plot below. |
| Systems of equations: | Two or more equations that are conditions imposed simultaneously on all the variables, but may or may not have common solutions (e.g., x + y = 2, and 3x + 2y = 5). |
| Symmetry: | A line of symmetry separates a figure into two congruent halves, each of which is a reflection of the other (e.g., , the line through the center of the circle divides it into congruent halves). |
| t-test: | A statistical test done to test the difference of means of two samples. |
| Tessellation: | A repetitive pattern of polygons that covers an area with no holes and no overlaps, like floor tiles. |
| Transformation: | An operation on a geometric figure by which each point gives rise to a unique image. |
| Translation: | A transformation that moves a geometric figure by sliding each of the points the same distance in the same direction. |
| Tree diagram: | A diagram used to show the total number of possible outcomes in a probability experiment. |
Trigonometric
functions: | A function (e.g., sine, cosine, tangent, cotangent, secant, cosecant) whose independent variable is an angle measure, usually in degrees or radians. |
| Valid argument: | An argument with the property no matter what statements are substituted in the premises, the truth value of the form is true. If the premises are true, then the conclusion is true. |
| Variable: | A symbol used to stand for any one of a given set of numbers or other objects (e.g., in the equation y = x + 5, y and x are variables). |
| Variance: | In a data set, the sum of the squared deviations divided by one less than the number of elements in the set (sample variance s2) or by the number of elements in the set (population variance phi2). |
| Vector: | A quantity that has both magnitude and direction (e.g., physical quantities such as velocity and force). |
| Venn diagram: | A display that pictures unions and intersections of sets. |
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