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## Content Area: MathematicsGrade Level Expectations: Sixth GradeStandard: 2. Patterns, Functions, and Algebraic Structures

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Algebraic expressions can be used to generalize properties of arithmetic Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Write and evaluate numerical expressions involving whole-number exponents. (CCSS: 6.EE.1) Write, read, and evaluate expressions in which letters stand for numbers. (CCSS: 6.EE.2)Write expressions that record operations with numbers and with letters standing for numbers.1 (CCSS: 6.EE.2a)Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient) and describe one or more parts of an expression as a single entity.2 (CCSS: 6.EE.2b)Evaluate expressions at specific values of their variables including expressions that arise from formulas used in real-world problems.3 (CCSS: 6.EE.2c)Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). (CCSS: 6.EE.2c) Apply the properties of operations to generate equivalent expressions.4 (CCSS: 6.EE.3) Identify when two expressions are equivalent.5 (CCSS: 6.EE.4) Inquiry Questions: If we didn’t have variables, what would we use? What purposes do variable expressions serve? What are some advantages to being able to describe a pattern using variables? Why does the order of operations exist? What other tasks/processes require the use of a strict order of steps? Relevance & Application: The simplification of algebraic expressions allows one to communicate mathematics efficiently for use in a variety of contexts. Using algebraic expressions we can efficiently expand and describe patterns in spreadsheets or other technologies. Nature Of: Mathematics can be used to show that things that seem complex can be broken into simple patterns and relationships. Mathematics can be expressed in a variety of formats. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians look for and make use of structure. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

1 For example, express the calculation "Subtract y from 5" as 5 – y. (CCSS: 6.EE.2a)

2 For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. (CCSS: 6.EE.2b)

3 For example, use the formulas $V = s^3$ and $A = 6s^2$ to find the volume and surface area of a cube with sides of length s = 1/2. (CCSS: 6.EE.2c)

4 For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. (CCSS: 6.EE.3)

5 i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. (CCSS: 6.EE.4)

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Variables are used to represent unknown quantities within equations and inequalities Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Describe solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? (CCSS: 6.EE.5) Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (CCSS: 6.EE.5) Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. (CCSS: 6.EE.6)Recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. (CCSS: 6.EE.6) Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. (CCSS: 6.EE.7) Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. (CCSS: 6.EE.8) Show that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (CCSS: 6.EE.8) Represent and analyze quantitative relationships between dependent and independent variables. (CCSS: 6.EE)Use variables to represent two quantities in a real-world problem that change in relationship to one another. (CCSS: 6.EE.9)Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. (CCSS: 6.EE.9)Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.6 (CCSS: 6.EE.9) Inquiry Questions: Do all equations have exactly one unique solution? Why? How can you determine if a variable is independent or dependent? Relevance & Application: Variables allow communication of big ideas with very few symbols. For example, $d = r \times t$ is a simple way of showing the relationship between the distance one travels and the rate of speed and time traveled, and $C = \pi d$ expresses the relationship between circumference and diameter of a circle. Variables show what parts of an expression may change compared to those parts that are fixed or constant. For example, the price of an item may be fixed in an expression, but the number of items purchased may change. Nature Of: Mathematicians use graphs and equations to represent relationships among variables. They use multiple representations to gain insights into the relationships between variables. Mathematicians can think both forward and backward through a problem. An equation is like the end of a story about what happened to a variable. By reading the story backward, and undoing each step, mathematicians can find the value of the variable. Mathematicians model with mathematics. (MP)

6 For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. (CCSS: 6.EE.9)

## Content Area: MathematicsGrade Level Expectations: Fourth GradeStandard: 2. Patterns, Functions, and Algebraic Structures

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Number patterns and relationships can be represented by symbols Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Generate and analyze patterns and identify apparent features of the pattern that were not explicit in the rule itself.1 (CCSS: 4.OA.5)Use number relationships to find the missing number in a sequenceUse a symbol to represent and find an unknown quantity in a problem situationComplete input/output tablesFind the unknown in simple equations Apply concepts of squares, primes, composites, factors, and multiples to solve problemsFind all factor pairs for a whole number in the range 1–100. (CCSS: 4.OA.4)Recognize that a whole number is a multiple of each of its factors. (CCSS: 4.OA.4)Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. (CCSS: 4.OA.4)Determine whether a given whole number in the range 1–100 is prime or composite. (CCSS: 4.OA.4) Inquiry Questions: What characteristics can be used to classify numbers into different groups? How can we predict the next element in a pattern? Why do we use symbols to represent missing numbers? Why is finding an unknown quantity important? Relevance & Application: Use of an input/output table helps to make predictions in everyday contexts such as the number of beads needed to make multiple bracelets or number of inches of expected growth. Symbols help to represent situations from everyday life with simple equations such as finding how much additional money is needed to buy a skateboard, determining the number of players missing from a soccer team, or calculating the number of students absent from school. Comprehension of the relationships between primes, composites, multiples, and factors develop number sense. The relationships are used to simplify computations with large numbers, algebraic expressions, and division problems, and to find common denominators. Nature Of: Mathematics involves pattern seeking. Mathematicians use patterns to simplify calculations. Mathematicians model with mathematics. (MP)

1 For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. (CCSS: 4.OA.5)

## Content Area: MathematicsGrade Level Expectations: High SchoolStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. Objects in the plane can be described and analyzed algebraically Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Express Geometric Properties with Equations. (CCSS: G-GPE)Translate between the geometric description and the equation for a conic section. (CCSS: G-GPE)Derive the equation of a circle of given center and radius using the Pythagorean Theorem. (CCSS: G-GPE.1)Complete the square to find the center and radius of a circle given by an equation. (CCSS: G-GPE.1)Derive the equation of a parabola given a focus and directrix. (CCSS: G-GPE.2)Use coordinates to prove simple geometric theorems algebraically. (CCSS: G-GPE)Use coordinates to prove simple geometric theorems11 algebraically. (CCSS: G-GPE.4)Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.12 (CCSS: G-GPE.5)Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (CCSS: G-GPE.6)Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.* (CCSS: G-GPE.7) Inquiry Questions: What does it mean for two lines to be parallel? What happens to the coordinates of the vertices of shapes when different transformations are applied in the plane? Relevance & Application: Knowledge of right triangle trigonometry allows modeling and application of angle and distance relationships such as surveying land boundaries, shadow problems, angles in a truss, and the design of structures. Nature Of: Geometry involves the investigation of invariants. Geometers examine how some things stay the same while other parts change to analyze situations and solve problems. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP)

11 For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, $\sqrt{3}$) lies on the circle centered at the origin and containing the point (0, 2). (CCSS: G-GPE.4)

12 e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point. (CCSS: G-GPE.5)

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 4. Attributes of two- and three-dimensional objects are measurable and can be quantified Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Explain volume formulas and use them to solve problems. (CCSS: G-GMD)Give an informal argument13 for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. (CCSS: G-GMD.1)Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* (CCSS: G-GMD.3) Visualize relationships between two-dimensional and three-dimensional objects. (CCSS: G-GMD)Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (CCSS: G-GMD.4) Inquiry Questions: How might surface area and volume be used to explain biological differences in animals? How is the area of an irregular shape measured? How can surface area be minimized while maximizing volume? Relevance & Application: Understanding areas and volume enables design and building. For example, a container that maximizes volume and minimizes surface area will reduce costs and increase efficiency. Understanding area helps to decorate a room, or create a blueprint for a new building. Nature Of: Mathematicians use geometry to model the physical world. Studying properties and relationships of geometric objects provides insights in to the physical world that would otherwise be hidden. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians model with mathematics. (MP)

13 Use dissection arguments, Cavalieri's principle, and informal limit arguments. (CCSS: G-GMD.1)

## Content Area: MathematicsGrade Level Expectations: Sixth GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Objects in space and their parts and attributes can be measured and analyzed Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Develop and apply formulas and procedures for area of plane figuresFind the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. (CCSS: 6.G.1)Apply these techniques in the context of solving real-world and mathematical problems. (CCSS: 6.G.1) Develop and apply formulas and procedures for volume of regular prisms.Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths. (CCSS: 6.G.2)Show that volume is the same as multiplying the edge lengths of a rectangular prism. (CCSS: 6.G.2)Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (CCSS: 6.G.2) Draw polygons in the coordinate plan to solve real-world and mathematical problems. (CCSS: 6.G.3)Draw polygons in the coordinate plane given coordinates for the vertices.Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. (CCSS: 6.G.3) Develop and apply formulas and procedures for the surface area.Represent three-dimensional figures using nets made up of rectangles and triangles. (CCSS: 6.G.4)Use nets to find the surface area of figures. (CCSS: 6.G.4)Apply techniques for finding surface area in the context of solving real-world and mathematical problems. (CCSS: 6.G.4) Inquiry Questions: Can two shapes have the same volume but different surface areas? Why? Can two figures have the same surface area but different volumes? Why? What does area tell you about a figure? What properties affect the area of figures? Relevance & Application: Knowledge of how to find the areas of different shapes helps do projects in the home and community. For example how to use the correct amount of fertilizer in a garden, buy the correct amount of paint, or buy the right amount of material for a construction project. The application of area measurement of different shapes aids with everyday tasks such as buying carpeting, determining watershed by a center pivot irrigation system, finding the number of gallons of paint needed to paint a room, decomposing a floor plan, or designing landscapes. Nature Of: Mathematicians realize that measurement always involves a certain degree of error. Mathematicians create visual representations of problems and ideas that reveal relationships and meaning. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians reason abstractly and quantitatively. (MP)

## Content Area: MathematicsGrade Level Expectations: Fifth GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Geometric figures can be described by their attributes and specific locations in the plane Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Graph points on the coordinate plane2 to solve real-world and mathematical problems. (CCSS: 5.G) Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. (CCSS: 5.G.2) Classify two-dimensional figures into categories based on their properties. (CCSS: 5.G)Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.3 (CCSS: 5.G.3)Classify two-dimensional figures in a hierarchy based on properties. (CCSS: 5.G.4) Inquiry Questions: How does using a coordinate grid help us solve real world problems? What are the ways to compare and classify geometric figures? Why do we classify shapes? Relevance & Application: The coordinate grid is a basic example of a system for mapping relative locations of objects. It provides a basis for understanding latitude and longitude, GPS coordinates, and all kinds of geographic maps. Symmetry is used to analyze features of complex systems and to create worlds of art. For example symmetry is found in living organisms, the art of MC Escher, and the design of tile patterns, and wallpaper. Nature Of: Geometry’s attributes give the mind the right tools to consider the world around us. Mathematicians model with mathematics. (MP) Mathematicians look for and make use of structure. (MP)

2 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. (CCSS: 5.G.1)
Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). (CCSS: 5.G.1)

3 For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. (CCSS: 5.G.3)

## Content Area: MathematicsGrade Level Expectations: Fourth GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 2. Geometric figures in the plane and in space are described and analyzed by their attributes Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. (CCSS: 4.G.1) Identify points, line segments, angles, and perpendicular and parallel lines in two-dimensional figures. (CCSS: 4.G.1) Classify and identify two-dimensional figures according to attributes of line relationships or angle size.6 (CCSS: 4.G.2) Identify a line of symmetry for a two-dimensional figure.7 (CCSS: 4.G.3) Inquiry Questions: How do geometric relationships help us solve problems? Is a square still a square if it’s tilted on its side? How are three-dimensional shapes different from two-dimensional shapes? What would life be like in a two-dimensional world? Why is it helpful to classify things like angles or shapes? Relevance & Application: The understanding and use of spatial relationships helps to predict the result of motions such as how articles can be laid out in a newspaper, what a room will look like if the furniture is rearranged, or knowing whether a door can still be opened if a refrigerator is repositioned. The application of spatial relationships of parallel and perpendicular lines aid in creation and building. For example, hanging a picture to be level, building windows that are square, or sewing a straight seam. Nature Of: Geometry is a system that can be used to model the world around us or to model imaginary worlds. Mathematicians look for and make use of structure. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

6 Based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. (CCSS: 4.G.2)

7 as a line across the figure such that the figure can be folded along the line into matching parts. (CCSS: 4.G.3)
Identify line-symmetric figures and draw lines of symmetry. (CCSS: 4.G.3)

## Content Area: MathematicsGrade Level Expectations: Third GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Geometric figures are described by their attributes Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Reason with shapes and their attributes. (CCSS: 3.G)Explain that shapes in different categories1 may share attributes2 and that the shared attributes can define a larger category.3 (CCSS: 3.G.1)Identify rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. (CCSS: 3.G.1)Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.4 (CCSS: 3.G.2) Inquiry Questions: What words in geometry are also used in daily life? Why can different geometric terms be used to name the same shape? Relevance & Application: Recognition of geometric shapes allows people to describe and change their surroundings such as creating a work of art using geometric shapes, or design a pattern to decorate. Nature Of: Mathematicians use clear definitions in discussions with others and in their own reasoning. Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians look for and make use of structure. (MP)

1 e.g., rhombuses, rectangles, and others. (CCSS: 3.G.1)

2 e.g., having four sides. (CCSS: 3.G.1)

4 For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. (CCSS: 3.G.2)

## Content Area: MathematicsGrade Level Expectations: First GradeStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Shapes can be described by defining attributes and created by composing and decomposing Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Distinguish between defining attributes1 versus non-defining attributes.2 (CCSS: 1.G.1) Build and draw shapes to possess defining attributes. (CCSS: 1.G.1) Compose two-dimensional shapes3 or three-dimensional shapes4 to create a composite shape, and compose new shapes from the composite shape. (CCSS: 1.G.2) Partition circles and rectangles into two and four equal shares. (CCSS: 1.G.3)Describe shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. (CCSS: 1.G.3)Describe the whole as two of, or four of the equal shares.5 (CCSS: 1.G.3) Inquiry Questions: What shapes can be combined to create a square? What shapes can be combined to create a circle? Relevance & Application: Many objects in the world can be described using geometric shapes and relationships such as architecture, objects in your home, and things in the natural world. Geometry gives us the language to describe these objects. Representation of ideas through drawing is an important form of communication. Some ideas are easier to communicate through pictures than through words such as the idea of a circle, or an idea for the design of a couch. Nature Of: Geometers use shapes to represent the similarity and difference of objects. Mathematicians model with mathematics. (MP) Mathematicians look for and make use of structure. (MP)

1 e.g., triangles are closed and three-sided. (CCSS: 1.G.1)

2 e.g., color, orientation, overall size. (CCSS: 1.G.1)

3 rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles. (CCSS: 1.G.2)

4 cubes, right rectangular prisms, right circular cones, and right circular cylinders. (CCSS: 1.G.2)

5 Understand for these examples that decomposing into more equal shares creates smaller shares. (CCSS: 1.G.3)

## Content Area: MathematicsGrade Level Expectations: KindergartenStandard: 4. Shape, Dimension, and Geometric Relationships

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Shapes can be described by characteristics and position and created by composing and decomposing Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). (CCSS: K.G)Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. (CCSS: K.G.1)Correctly name shapes regardless of their orientations or overall size. (CCSS: K.G.2)Identify shapes as two-dimensional1 or three dimensional.2 (CCSS: K.G.3) Analyze, compare, create, and compose shapes. (CCSS: K.G)Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts3 and other attributes.4 (CCSS: K.G.4)Model shapes in the world by building shapes from components5 and drawing shapes. (CCSS: K.G.5)Compose simple shapes to form larger shapes.6 (CCSS: K.G.6) Inquiry Questions: What are the ways to describe where an object is? What are all the things you can think of that are round? What is the same about these things? How are these shapes alike and how are they different? Can you make one shape with other shapes? Relevance & Application: Shapes help people describe the world. For example, a box is a cube, the Sun looks like a circle, and the side of a dresser looks like a rectangle. People communicate where things are by their location in space using words like next to, below, or between. Nature Of: Geometry helps discriminate one characteristic from another. Geometry clarifies relationships between and among different objects. Mathematicians model with mathematics. (MP) Mathematicians look for and make use of structure. (MP)

1 lying in a plane, "flat". (CCSS: K.G.3)

2 "solid". (CCSS: K.G.3)

3 e.g., number of sides and vertices/"corners". (CCSS: K.G.4)

4 e.g., having sides of equal length. (CCSS: K.G.4)

5 e.g., sticks and clay balls. (CCSS: K.G.5)

6 For example, "Can you join these two triangles with full sides touching to make a rectangle?" (CCSS: K.G.6)