# BPS District Mathematics Standards Book

Completion requirements

### Math Domains/Progressions

#### Geometry and Measurement

#### (G) Geometry K-5

*Learners will compose and classify figures and shapes based on attributes and properties; represent and solve problems using a coordinate plane.*

#### (GF) Geometric Figures MS

*Learners will use visualization, spatial reasoning, and geometric modeling to investigate the characteristics of figures, perform transformations, and construct logical arguments.*

** Two-Dimensional Shapes**

**MAT-00.GM.G.01**Name shapes and identify them as two-dimensional (squares, circles, triangles, rectangles)regardless of their orientations or overall size.**MAT-00.GM.G.03**Compare and classify two-dimensional shapes to describe their similarities, differences, and attributes (squares, circles, triangles, rectangles).**MAT-01.GM.G.01**Name shapes and identify them as two-dimensional (trapezoids, rhombuses, pentagons, hexagons, octagons).**MAT-01.GM.G.03**Determine geometric attributes of two-dimensional and three-dimensional shapes.**MAT-02.GM.G.01**Identify two-dimensional shapes (parallelograms and quadrilaterals).**MAT-02.GM.G.03**Compose geometric shapes having specified geometric attributes, such as a given number of edges, angles, faces, vertices, and/or sides.**MAT-03.GM.G.01**In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.**MAT-03.GM.G.02**Sort quadrilaterals into categories based on attributes.**MAT-04.GM.G.01**Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).**MAT-04.GM.G.02**Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of specified size.**MAT-05.GM.G.01**Classify two-dimensional figures in a hierarchy based on properties.**MAT-10.GM.01**Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and plane.**MAT-10.GM.09**Prove and apply theorems about lines and angles.**MAT-10.GM.10**Prove and apply theorems about triangles.**MAT-10.GM.11**Prove and apply theorems about parallelograms.**MAT-10.GM.34**Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

** Three-Dimensional Shapes**

**MAT-00.GM.G.02**Name shapes and identify them as three-dimensional (cubes and spheres) regardless of their orientations or overall size.**MAT-01.GM.G.02**Name and identify solids as three-dimensional (cylinders, cones, triangular prisms, and rectangular prisms).**MAT-01.GM.G.03**Determine geometric attributes of two-dimensional and three-dimensional shapes.**MAT-02.GM.G.02**Identify two-dimensional shapes found within three-dimensional shapes.**MAT-02.GM.G.03**Compose geometric shapes having specified geometric attributes, such as a given number of edges, angles, faces, vertices, and/or sides.**MAT-06.GM.GF.03**Represent three-dimensional figures using nets made up of rectangles and triangles (right prisms and pyramids whose bases are triangles and rectangles). Calculate the surface area of prisms with rectangular and triangular bases using nets, including authentic problems.

** Compose Shapes**

**MAT-00.GM.G.04**Compose a geometric shape by combining two or more simple shapes.**MAT-01.GM.G.04**Compose a geometric shape or solid by combining multiple two-dimensional shapes and/or three-dimensional solids.**MAT-02.GM.G.03**Compose shapes having specified geometric attributes, such as a given number of edges, angles, faces, vertices, and/or sides.**MAT-07.GM.GF.01**Draw triangles from given conditions using appropriate tools. Defend whether a unique triangle, multiple triangles, or no triangle can be constructed when given three measures of angles or sides.**MAT-10.GM.12**Make basic geometric constructions (e.g., segments, angles, bisectors, parallel and perpendicular lines) with a variety of tools and methods.**MAT-10.GM.13**Apply basic construction to create polygons such as equilateral triangles, squares, and regular hexagons inscribed in a circle.

** Symmetry**

**MAT-03.GM.G.03**Identify lines of symmetry in quadrilaterals.**MAT-04.GM.G.03**Draw lines of symmetry in two-dimensional figures

** Coordinate Plane**

**MAT-03.GM.G.01**In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.**MAT-05.GM.G.02**Identify the x-coordinate and y-coordinate to graph and name points in the first quadrant of the coordinate plane.**MAT-05.GM.G.03**Form ordered pairs and graph points in the first quadrant of the coordinate plane to solve authentic word problems.**MAT-06.GM.GF.01**Identify and position ordered pairs of rational numbers in all four quadrants of a coordinate plane.**MAT-06.GM.GF.02**Draw polygons in the coordinate plane given coordinates for vertices. Determine the length of a side joining points with the same first or second coordinate, including authentic problems.**MAT-10.GM.27**Develop and verify the slope criteria for parallel and perpendicular lines. Apply the slope criteria for parallel and perpendicular lines to solve geometric problems using algebra.**MAT-10.GM.28**Verify simple geometric theorems algebraically using coordinates. Verify algebraically, using coordinates, that a given set of points produces a particular type of triangle or quadrilateral.**MAT-10.GM.29**Determine the midpoint or endpoint of a line segment using coordinates. (+) Find the point on a directed line segment between two given points that partitions the segments in a given ratio.**MAT-12.NO.10**Represent complex numbers on the complex plane in rectangular, trigonometric, and polar forms. Find the modulus (absolute value) of a complex number. Explain why the rectangular, trigonometric, and polar forms of a given complex number represent the same number.**MAT-12.NO.11**Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.**MAT-12.NO.14**Recognize vector quantities as having both magnitude and direction, writing them in polar form.**MAT-12.NO.15**Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.**MAT-12.NO.16**Solve problems involving magnitude and direction that can be represented by vectors.**MAT-12.NO.17**Add and subtract vectors.**MAT-12.NO.18**Multiply a vector by a scalar.**MAT-12.AR.F.18**Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.**MAT-12.AR.F.19**Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.

** Transformations**

**MAT-08.GM.GF.01**Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.**MAT-08.GM.GF.02**Describe the characteristics of transformations on the coordinate plane using transformation language.**MAT-08.GM.GF.03**Name the type of transformation(s) needed to map a pre-image to its image.**MAT-09.AR.F.09**Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative). Find the values of a, h, and k given the graph of the function.**MAT-10.GM.02**Represent transformations in the plane. Describe transformations as functions that take points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).**MAT-10.GM.03**Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.**MAT-10.GM.04**Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.**MAT-10.GM.05**Draw the image of a figure that has undergone a series of transformations [rotation(s),- reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).
**MAT-10.GM.06**Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.**MAT-10.GM.14**Verify experimentally and justify the properties of dilations given by a center and a scale factor.**MAT-10.GM.15**Use transformations to decide if two given figures are similar. Apply the meaning of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.**MAT-10.GM.16**Prove similarity theorems about triangles.**MAT-10.GM.17**Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.**MAT-12.AR.F.4**Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x-h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.**MAT-12.NO.18**Multiply a vector by a scalar.

** Congruence and Similarity**

**MAT-08.GM.GF.01**Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.**MAT-08.GM.GF.02**Describe the characteristics of transformations on the coordinate plane using transformation language.**MAT-10.GM.07**Use the definition of congruence, based on rigid motions, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.**MAT-10.GM.08**Prove two triangles are congruent using the congruence theorems.**MAT-10.GM.17**Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.

** Angles/Triangles**

**MAT-03.GM.G.01**In two-dimensional shapes, identify lines, angles (right, acute, obtuse), and perpendicular and parallel lines.**MAT-04.GM.G.01**Identify, label, and draw points, lines, line segments, rays, and angles (right, acute, obtuse).**MAT-04.GM.M.07**Recognize angle measures as additive and solve addition and subtraction problems to find unknown angles on a diagram.**MAT-07.GM.GF.01**Draw triangles from given conditions using appropriate tools. Defend whether a unique triangle, multiple triangles, or no triangle can be constructed when given three measures of angles or sides.**MAT-07.GM.GF.02**Describe the angle-pair relationships: supplementary angles, complementary angles, vertical angles, and adjacent angles. Solve for an unknown angle in a figure by applying facts about these angles.**MAT-08.GM.GF.04**Describe the following angle-pair relationships: interior and exterior angles of triangles and angles formed when a transversal cuts parallel lines or intersecting lines. Solve for an unknown angle in a figure by applying facts about these angles.**MAT-08.GM.GF.05**Describe the relationship between the leg length and the hypotenuse length of a right triangle. Determine whether a triangle is a right triangle using this relationship.**MAT-08.GM.GF.06**Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in two and three dimensions on and off a coordinate plane, including authentic problems.**MAT-10.GM.01**Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, and plane.**MAT-10.GM.09**Prove and apply theorems about lines and angles.**MAT-10.GM.10**Prove and apply theorems about triangles.**MAT-10.GM.18**Recognize how the properties of similar right triangles allow for trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.**MAT-10.GM.19**Explain and use the relationship between the sine and cosine of complementary angles.**MAT-10.GM.20**Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30º, -60º, -90º, and 45º-45º-90º).**MAT-10.GM.21**Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.**MAT-10.GM.23**Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.**MAT-10.GM.24**Construct a tangent line from a point outside a given circle to the circle.**MAT-10.GM.26**Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.**MAT-12.AR.F.16**Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π/3, π/4, and π/6.**MAT-12.AR.F.17**Use the Pythagorean Identity sin² (θ) + cos² (θ) = 1 to find sin (θ), cos (θ), or tan (θ) given sin (θ), cos (θ), or tan (θ) and the quadrant of the angle.**MAT-12.GM.03**Determine and apply appropriate formulas to solve right and non-right triangle problems in context

#### (M) Measurement K-5

*Learners will represent and calculate measurement data, including time, money, and geometric measurement, and convert like measurement units within a given system.*

#### (AV) Area and Volume MS

*Learners will use visualization and spatial reasoning to solve problems involving the area, surface area, and volume of geometric figures.*

** Measure Length**

**MAT-01.GM.M.01**Measure the length of an object as a whole number of same-size, non-standard units from end to end.**MAT-02.GM.M.01**Measure the length of an object using two different standard units of measurement. Describe how the two measurements relate to the size of the units chosen.**MAT-03.GM.M.01**Measure lengths using rulers marked with halves and fourths of an inch.**MAT-10.GM.25**Explain and use the formulas for arc length and area of sectors of circles.**MAT-10.GM.26**Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.

** Compare Objects**

**MAT-00.GM.M.01**Compare and order two objects with a common measurable attribute.**MAT-01.GM.M.02**Compare the lengths of three objects using a common measurable attribute.**MAT-02.GM.M.02**Estimate and measure to determine how much longer one object is than another, expressing the difference with a standard unit of measurement.

** Units of Measurement**

**MAT-03.GM.M.02**Measure and estimate liquid volumes and masses of objects using standard units. Solve one step authentic word problems involving masses or volumes given in the same units.**MAT-04.GM.M.01**Know the relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec. Record measurement equivalents in a two-column table.**MAT-04.GM.M.03**Identify and use the appropriate tools, operations, and units of measurement, both customary and metric to solve problems involving time, length, weight, mass, and capacity.**MAT-04.GM.M.02**Generate simple conversions from a larger unit to a smaller unit to solve authentic problems within a single system of measurement, both customary and metric systems.**MAT-05.GM.M.01**Generate conversions among different-sized standard measurement units within a given measurement system, both customary and metric systems. Use these conversions in solving multi-step, authentic word problems.**MAT-06.AR.RP.05**Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.**MAT-10.GM.36**Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).**MAT-10.NO.03**Choose and interpret the scale and the origin in graphs and data displays.**MAT-10.NO.04**Define appropriate quantities and units for the purpose of descriptive modeling.**MAT-10.NO.05**Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.**MAT-12.NO.04**Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.**MAT-12.NO.05**Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

** Time**

**MAT-00.GM.M.02**Tell time as it relates to daily life (today, yesterday, tomorrow, morning, afternoon, night).**MAT-01.GM.M.03**Tell and write time to the hour and half-hour (including o'clock and half past) using analog and digital clocks.**MAT-02.GM.M.03**Tell and write time to the nearest five minutes (including quarter after and quarter to) with a.m. and p.m. using analog and digital clocks.**MAT-03.GM.M.03**Tell and write time to the nearest minute and measure time intervals in minutes.**MAT-03.GM.M.04**Solve elapsed time authentic word problems on the hour and the half-hour, using a variety of strategies.

** Money**

**MAT-01.GM.M.04**Identify and tell the value of a dollar bill, quarter, dime, nickel, and penny.**MAT-01.GM.M.05**Count collections of coins (pennies, nickels, and dimes) relating to patterns of counting by 1s, 5s, and 10s up to one dollar.**MAT-02.GM.M.04**Count collections of money (quarters, dimes, nickels, and pennies) relating to patterns of counting by 1s, 5s, and 10s up to one dollar.**MAT-03.GM.M.05**Solve authentic word problems involving dollar bills, quarters, dimes, nickels, and pennies using the $ and ¢ symbols appropriately.**MAT-04.GM.M.04**Solve word authentic problems involving dollar bills, quarters, dimes, nickels, and pennies using the $ and ¢ symbols and decimal notation appropriately.

** Perimeter**

**MAT-03.GM.M.06**Solve problems involving the perimeters of rectangles given the side lengths or when given the perimeter and unknown side length(s).**MAT-04.GM.M.05**Apply the area and perimeter formulas for rectangles, including connected rectangular figures, in problems.**MAT-05.GM.M.02**Find the area and perimeter of a rectangle, including connected rectangular figures, with fractional side lengths.**MAT-07.GM.AV.01**Describe the relationship between the circumference and diameter of a circle (pi). Apply given formulas to calculate the area and circumference of a circle, including in authentic problems.**MAT-10.GM.30**Compute perimeters of polygons and areas of triangles, parallelograms, trapezoids, and kites using coordinates.

** Area/Surface Area**

**MAT-03.GM.M.07**Recognize area as an attribute of plane figures and understand concepts of area measurement.**MAT-03.GM.M.08**Find the area of a rectangle with whole-number side lengths by modeling with unit squares; show that area can be additive and is the same as found by multiplying the side lengths.**MAT-04.GM.M.05**Apply the area and perimeter formulas for rectangles, including connected rectangular figures, in problems.**MAT-05.GM.M.02**Find the area and perimeter of a rectangle, including connected rectangular figures, with fractional side lengths.**MAT-06.GM.AV.01**Derive the relationship of the areas of triangles using the area of rectangles. Calculate the areas of triangles and quadrilaterals in authentic and mathematical problems by composing and/or decomposing them into rectangles and triangles.**MAT-06.GM.GF.03**Represent three-dimensional figures using nets made up of rectangles and triangles (right prisms and pyramids whose bases are triangles and rectangles). Calculate the surface area of prisms with rectangular and triangular bases using nets. Apply these techniques in the context of solving authentic and mathematical problems.**MAT-07.GM.AV.02**Calculate areas of polygons by composing and/or decomposing them into rectangles and triangles, including in authentic problems. Solve problems involving the surface area of prisms and right pyramids using nets, including authentic problems.**MAT-10.GM.30**Compute perimeters of polygons and areas of triangles, parallelograms, trapezoids, and kites using coordinates.**MAT-12.GM.04**Derive the formula A=1 ab sin C for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

** Volume**

**MAT-05.GM.M.03**Recognize volume as an attribute of rectangular prisms and measure volume by counting unit cubes.**MAT-06.GM.AV.02**Describe the concept of volume of a right rectangular prism. Apply given formulas to calculate the volume of right rectangular prisms, including fractional edge lengths, including authentic problems.**MAT-07.GM.AV.03**Solve problems involving the volume of prisms and composite solids, including authentic problems.**MAT-08.GM.AV.01**Apply given formulas to solve problems involving the volume of cones, cylinders, and spheres, including authentic problems.**MAT-10.GM.32**Calculate the surface area for prisms, cylinders, pyramids, cones, and spheres to solve problems.**MAT-10.GM.33**Know and apply volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.**MAT-10.GM.35**Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

** Circle Measurements**

**MAT-07.GM.AV.01**Describe the relationship between the circumference and diameter of a circle (pi). Apply the given formula to calculate the area and circumference of a circle, including in authentic problems.**MAT-10.GM.22**Apply theorems about relationships between line segments and circles or angles and circles formed by radii, diameter, secants, tangents, and chords to find unknown lengths or angles.**MAT-10.GM.25**Explain and use the formulas for arc length and area of sectors of circles.**MAT-10.GM.26**Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.**MAT-10.GM.31**Explain derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone.**MAT-12.GM.01**Write the equation of a conic section given its special features. Convert between the standard form and general form equations of conic sections.**MAT-12.GM.02**Identify key features of a conic section given its equation. Apply properties of conic sections in context.**MAT-12.AR.F.18**Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.