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EAST LONGMEADOW PUBLIC SCHOOLS Mathematics The Massachusetts Mathematics Curriculum Framework envisions all students in the Commonwealth achieving mathematical competence through a strong mathematics program that emphasizes problem solving, communicating, reasoning and proof, making connections, and using representations. Acquiring such competence depends in large part on a clear, comprehensive, coherent, and developmentally appropriate set of standards to guide curriculum expectations. |
| Geometry |
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Problem Solving: Transformations
The learner will be able to apply concepts of the different types of transformations in solving problems.
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Geometric Concepts: Apply
The learner will be able to apply the concepts of vectors, phase shift, maxima, minima, points of inflection, and/or exact mathematical descriptions of symmetry to find and/or relate objects and their orientation.
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Angles/Triangles
The learner will be able to calculate the length of the missing side of a given triangle, know both the meaning and relationship between complementary and supplementary angles, and apply the triangle sum property to find the value of the third angle.
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Angles: Complementary/Supplementary
The learner will be able to identify, describe, and apply measures of complementary and supplementary angles.
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Angles: Parallel/Transversal
The learner will be able to identify angle relationships created by parallel and transversal lines.
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Angles: Properties/Apply
The learner will be able to apply properties of angles in solving both mathematical and real-world problems.
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Angles: Right/Real-World
The learner will be able to identify right angles within real-world situations.
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Angles: Solve Problems
The learner will be able to obtain solutions to problems with complementary, supplementary, and congruent angles, including vertical angles, angles formed when parallel lines are cut by a transversal, and angles in polygons.
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Circle Parts: Apply
The learner will be able to apply the properties of circle parts in real-world situations.
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Circle Parts: Arc/Area
The learner will be able to compute area of sectors of a circle.
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Circle Parts: Pi
The learner will be able to understand the relationship which pi represents.
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Circle: Arc/Chord/Angle
The learner will be able to apply knowledge of chords, vertical angles, arcs, and circle relationships to determine an unknown angle between intersecting chords.
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Circle: Diameter/Circumference Problems
The learner will be able to solve a problem in the context of a real world scenario with diameter and circumference.
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Circle: Real-World Problems
The learner will be able to solve real-world problems applying geometric properties of circles.
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Circle: Relating Diameter/Circumference
The learner will be able to determine how diameter and circumference are related.
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Circle: Relationships
The learner will be able to analyze how concentric, congruent, and similar circles are related.
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Circle: Relationships
The learner will be able to determine the relationships among the radius, diameter, chord, center, and circumference.
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Circles: Apply Concepts
The learner will be able to apply the various measurements associated with circles (pi, area, circumference, diameter, and radius) within practical contexts.
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Circles: Sphere/Problem Solving
The learner will be able to apply spheres to obtain solutions to problems associated with the definition of a sphere and sets of points related to the sphere.
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Congruence/Similarity: Apply
The learner will be able to apply the concepts and properties of congruence and similarity.
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Diagonals: Polygon/Draw
The learner will be able to draw diagonals of polygons.
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Diagonals: Rectangle/Story Problem
The learner will be able to determine the length of the diagonals of a rectangle within the context of a real world problem.
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Figures: Problem Solving/3-D Diagram
The learner will be able to construct a diagram to aid in the solution of a problem involving three-dimensional space.
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Geometric Concepts: Problem Solving
The learner will be able to solve problems with relationships between geometric concepts and real-world situations.
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Geometric Probability: Solving
The learner will be able to solve problems involving geometric probability by using areas.
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Inductive/Deductive: Conclusions
The learner will be able to form inductive and deductive conclusions within mathematical contexts.
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Line Segment: Midpoint
The learner will be able to determine the midpoint of a line segment.
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Logic: Converse/Inverse/Contrapositive
The learner will be able to identify the truth of an implication, its converse, inverse, and contrapositive.
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Logic: Problem Solving
The learner will be able to solve basic logic problems.
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Polygons: Problem Solving
The learner will be able to apply the properties, definitions, and/or theorems of polygons to obtain solutions to problems related to the interior and exterior angles of convex polygons.
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Properties: Right Triangles
The learner will be able to use the properties of right triangles to obtain problem solutions using relationships in special right triangles.
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Pythagorean Theorem
The learner will be able to use the Pythagorean Theorem to determine an unknown side length of a right triangle.
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Pythagorean Theorem: Analyze
The learner will be able to identify situations requiring the Pythagorean theorem, and analyze how and why the theorem is used.
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Pythagorean Theorem: Apply
The learner will be able to apply the Pythagorean Theorem.
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Pythagorean Theorem: Triples
The learner will be able to identify Pythagorean triples, such as 3-4-5 and 5-12-13, in right triangles.
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Quadrilaterals: Angle Sum
The learner will be able to understand that the sum of the angles of any quadrilateral equals 360 degrees.
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Solids: Problem Solving
The learner will be able to solve geometric problems by analyzing three-dimensional solids.
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Triangles: 30-60-90 Right/Identify
The learner will be able to identify the lengths of the missing sides of a 30-60-90 degree right triangle given the length of one side.
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Triangles: 45-45-90 Right/Identify
The learner will be able to identify the lengths of the missing sides of a 45-45-90 degree right triangle given the length of one side.
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Triangles: Angle Sum
The learner will be able to understand that the sum of the angles of any triangle equals 180 degrees.
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Triangles: Isosceles/Apply
The learner will be able to apply knowledge of isosceles triangles and the triangle sum theorem in solving for an unknown angle measure.
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Triangles: Isosceles/Sum Theorem
The learner will be able to apply knowledge of isosceles triangles and the triangle sum theorem to solve for the measure of a base angle in an isosceles triangle when the other base angle and the third angle are given in binomial equations of the first degree.
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Triangles: Longest Side
The learner will be able to identify the longest side of a triangle.
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Triangles: Problem Solving
The learner will be able to apply triangle concepts in solving real-world problems .
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| Number Theory |
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Math Structures: Math Systems
The learner will be able to show that apparently unlike mathematical systems can be similar or identical.
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Divisibility
The learner will be able to determine which number is evenly divisible by a given number.
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Ratio/Proportion: Apply
The learner will be able to apply ratio and proportion to everyday scenarios.
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Ratio/Proportion: Problem Solving
The learner will be able to obtain problem solutions through the use of ratios and/or proportions.
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Ratios/Percents/Decimals/Fractions
The learner will be able to write comparisons as ratios, percents, decimals, and fractions.
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Square Roots: Perfect Square/Applying
The learner will be able to identify the square root of numbers which are perfect squares when given in the context of a real-world scenario.
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| Probability/Statistics |
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Statistical Experiments
The learner will be able to study a problem, establish the information needed to solve the problem, devise an experiment, carry it out, collect the data, draw conclusions based on the data, and communicate the results to others.
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Predictions: Curve Fitting
The learner will be able to make predictions from data using curve fitting.
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Venn Diagrams: Apply
The learner will be able to apply Venn diagrams that use two or three overlapping categories to obtain solutions to logic problems.
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Average
The learner will be able to understand the effect of additional data on the average.
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Average: Weighted Average
The learner will be able to solve problems that involve the calculation of a weighted average.
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Chance: Apply
The learner will be able to apply concepts of chance.
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Combinations/Permutations: Illustrate
The learner will be able to illustrate all permutations and combinations four items or less.
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Complementary Events: Probability
The learner will be able to calculate the probability of complementary events.
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Counting Methods: Apply
The learner will be able to apply counting methods.
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Counting Methods: Problem Solving
The learner will be able to solve real-world problems by applying counting methods.
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| Trigonometry |
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Triangles: Problem Solving
The learner will be able to solve real-world triangle problems using trigonometric concepts.
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Periodic Phenomena: Sine/Cosine
The learner will be able to investigate real-world phenomena which is periodic using the sine and cosine functions.
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Angle Measure: Converting
The learner will be able to convert between angle and radian measure.
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Angle Measure: Degree/Radian
The learner will be able to interchangeably use degrees and radians to illustrate angle measure and describe the advantages and/or disadvantages of their choice.
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Angle Measure: Degree/Radian
The learner will be able to comprehend the relationship between measure in radians and degrees.
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Angle Measure: Relating Degree/Radian
The learner will be able to determine the relationship between measure in radians and degrees.
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Angle Measures: Relating Degree/Radian
The learner will be able to evaluate how degrees and radians are related.
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Angles: Degrees/Equations
The learner will be able to solve an equation involving determining trigonometric values by substituting degree measures.
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Circle: Unit/Sine/Cosine
The learner will be able to comprehend the idea of sine and cosine being defined as the x and y coordinates of points on the unit circle.
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Connecting: Across Forms
The learner will be able to solve a given problem for the appropriate trigonometric ratios, radian measures, and angle measures.
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Connecting: Graphing/Trig. Functions
The learner will be able to use basic methods of graphic representation on trigonometric functions.
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Polar Coordinates: Define
The learner will be able to give a definition of polar coordinates.
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Polar Coordinates: Graph
The learner will be able to graph different points in the polar coordinate system.
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Polar Equations: Graph
The learner will be able to represent polar equations in graphical form.
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Triangle: Area/Trigonometric Properties
The learner will be able to solve for the area of a triangle using trigonometric properties.
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Triangles: Apply Trigonometry
The learner will be able to apply trigonometry in finding the unknown sides and/or angles of right triangles.
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Triangles: Develop Trigonometry Relation
The learner will be able to use a right triangle to develop the following trigonometric relationships: sine, cosine, and tangent.
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Triangles: Oblique/Problem Solving
The learner will be able to solve real-world oblique triangle problems using trigonometric concepts.
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Triangles: Pythagorean Triples
The learner will be able to use the relationships between Pythagorean triples to calculate the sine, cosine, and tangent values of angles.
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Triangles: Right/Problem Solving
The learner will be able to solve real-world right triangle problems using trigonometric concepts.
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Triangles: Solving General
The learner will be able to use trigonometric ratios to solve general triangles.
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Triangles: Solving Oblique
The learner will be able to use trigonometric ratios to solve oblique triangles.
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Triangles: Solving Right
The learner will be able to use trigonometric ratios to solve right triangles.
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Triangles: Solving Right
The learner will be able to solve right triangles.
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Triangles: Solving Right
The learner will be able to find the measure of each unknown side (to the nearest tenth or hundredth) or angle (to the nearest degree), solve real-world trigonometry problems, and include the angle of depression or elevation in calculations.
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Triangles: SSA Case
The learner will be able to discern whether 0,1, or 2 triangles exist, and what they are in an SSA (side, side, angle) case.
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Triangles: Trigonometric Ratios/30-60-90
The learner will be able to apply an understanding of the 30-60-90 Triangle Theorem to calculate the values of sine, cosine, and tangent of 30 and 60 degree angles.
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Triangles: Trigonometric Ratios/45-45-90
The learner will be able to apply an understanding of the 45-45-90 Triangle Theorem to calculate the values of sine, cosine, and tangent of a 45 degree angle.
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Triangles: Trigonometric Relationships
The learner will be able to apply trigonometric relationships in order to solve problems.
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Trig Concepts: Law of Sines and Cosines
The learner will be able to apply the Law of Sines and the Law of Cosines to solve oblique triangles.
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Trig. Equations: Amplitude
The learner will be able to identify the amplitude of a given trigonometric equation.
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Trig. Equations: Period
The learner will be able to identify the period of the graph of a given trigonometric equation.
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Trig. Equations: Solutions
The learner will be able to determine possible solutions for trigonometric equations.
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Trig. Identities
The learner will be able to identify and use the trigonometric identities.
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Trig. Ratios: Law of Sines/Cosines
The learner will be able to use the Law of Sines and Cosines to solve problems.
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Trig. Ratios: Law of Sines/Derive
The learner will be able to develop the Law of Sines by working with triangles, their angles, and their side lengths.
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Trig. Ratios: Negative Angle
The learner will be able to find the trigonometric value of a negative angle measure.
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Trig. Ratios: Positive/Negative
The learner will be able to use given knowledge of trigonometric values to determine in what quadrant an unknown angle lies.
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Trig. Ratios: Problem Solving
The learner will be able to use the trigonometric ratios of angle measure to solve problems.
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Trig. Ratios: Quadrants
The learner will be able to list the signs of each of the trigonometric equations in all four quadrants.
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Trig. Ratios: Quadrantal Angles
The learner will be able to determine the values of the trigonometric ratios of the quadrantal angles (0, 90, 180, etc.).
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Trig. Ratios: Sine/Cosine/Tangent
The learner will be able to determine the numerical values for the sine, cosine, and/or tangent of a given angle within a right triangle from an illustration where the side lengths are given.
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Trig. Ratios: Sine/Cosine/Tangent
The learner will be able to find the sine, cosine, and tangent given angle degrees.
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Trig. Ratios: Standard Position
The learner will be able to determine the trigonometric ratios of an angle in standard position.
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Trig. Ratios: Trigonometric Tables
The learner will be able to find either the sine, cosine, or tangent of right triangles by utilizing a table of trigonometric ratios.
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Trig. Ratios: Understand
The learner will be able to understand basic trigonometric ratios.
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Trigonometric Identities: Knowledge
The learner will be able to know the following trig identity: cosine squared + sine squared = 1.
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| Algebraic Concepts |
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Algebraic Concepts: Problem Solving
The learner will be able to use algebraic methods to solve real-world problems.
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