The Colorado Department of Education

Offices | Staff Contacts | Colorado.gov

New Colorado P-12 Academic Standards

Current Display Filter: Mathematics - All - by Specific Prepared Graduate Competency - (Remove PGC Filter)

Content Area: Mathematics
Grade Level Expectations: First Grade
Standard: 1. Number Sense, Properties, and Operations

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

2. Number relationships can be used to solve addition and subtraction problems
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Represent and solve problems involving addition and subtraction. (CCSS: 1.OA)
    • Use addition and subtraction within 20 to solve word problems.2 (CCSS: 1.OA.1)
    • Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.3 (CCSS: 1.OA.2)
  2. Apply properties of operations and the relationship between addition and subtraction. (CCSS: 1.OA)
    • Apply properties of operations as strategies to add and subtract.4 (CCSS: 1.OA.3)
    • Relate subtraction to unknown-addend problem.5 (CCSS: 1.OA.4)
  3. Add and subtract within 20. (CCSS: 1.OA)
    • Relate counting to addition and subtraction.6 (CCSS: 1.OA.5)
    • Add and subtract within 20 using multiple strategies.7 (CCSS: 1.OA.6)
    • Demonstrate fluency for addition and subtraction within 10. (CCSS: 1.OA.6)
  4. Use addition and subtraction equations to show number relationships. (CCSS: 1.OA)
    • Use the equal sign to demonstrate equality in number relationships.8 (CCSS: 1.OA.7)
    • Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.9 (CCSS: 1.OA.8)

Inquiry Questions:

  1. What is addition and how is it used?
  2. What is subtraction and how is it used?
  3. How are addition and subtraction related?

Relevance & Application:

  1. Addition and subtraction are used to model real-world situations such as computing saving or spending, finding the number of days until a special day, or determining an amount needed to earn a reward.
  2. Fluency with addition and subtraction facts helps to quickly find answers to important questions.

Nature Of:

  1. Mathematicians use addition and subtraction to take numbers apart and put them back together in order to understand number relationships.
  2. Mathematicians make sense of problems and persevere in solving them. (MP)
  3. Mathematicians look for and make use of structure. (MP)

2 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (CCSS: 1.OA.1)

3 e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (CCSS: 1.OA.2)

4 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.). (CCSS: 1.OA.3)

5 For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. (CCSS: 1.OA.4)

6 e.g., by counting on 2 to add 2. (CCSS: 1.OA.5)

7 Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 +7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). (CCSS: 1.OA.6)

8 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. (CCSS: 1.OA.7)

9 For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?. (CCSS: 1.OA.8)

Content Area: Mathematics
Grade Level Expectations: Kindergarten
Standard: 1. Number Sense, Properties, and Operations

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

2. Composing and decomposing quantity forms the foundation for addition and subtraction
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Model and describe addition as putting together and adding to, and subtraction as taking apart and taking from, using objects or drawings. (CCSS: K.OA)
    • Represent addition and subtraction with objects, fingers, mental images, drawings, sounds,6 acting out situations, verbal explanations, expressions, or equations. (CCSS: K.OA.1)
    • Solve addition and subtraction word problems, and add and subtract within 10.7 (CCSS: K.OA.2)
    • Decompose numbers less than or equal to 10 into pairs in more than one way.8 (CCSS: K.OA.3)
    • For any number from 1 to 9, find the number that makes 10 when added to the given number.9 (CCSS: K.OA.4)
    • Use objects including coins and drawings to model addition and subtraction problems to 10 (PFL)
  2. Fluently add and subtract within 5. (CCSS: K.OA.5)
  3. Compose and decompose numbers 11–19 to gain foundations for place value using objects and drawings.10 (CCSS: K.NBT)

Inquiry Questions:

  1. What happens when two quantities are combined?
  2. What happens when a set of objects is separated into different sets?

Relevance & Application:

  1. People combine quantities to find a total such as number of boys and girls in a classroom or coins for a purchase.
  2. People use subtraction to find what is left over such as coins left after a purchase, number of toys left after giving some away.

Nature Of:

  1. Mathematicians create models of problems that reveal relationships and meaning.
  2. Mathematics involves the creative use of imagination.
  3. Mathematicians reason abstractly and quantitatively. (MP)
  4. Mathematicians model with mathematics. (MP)

6 e.g., claps. (CCSS: K.OA.1)

7 e.g., by using objects or drawings to represent the problem. (CCSS: K.OA.2)

8 e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). (CCSS: K.OA.3)

9 e.g., by using objects or drawings, and record the answer with a drawing or equation. (CCSS: K.OA.4)

10 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. (CCSS: K.NBT.1)

Content Area: Mathematics
Grade Level Expectations: High School
Standard: 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 the plane can be transformed, and those transformations can be described and analyzed mathematically
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Experiment with transformations in the plane. (CCSS: G-CO)
    • State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (CCSS: G-CO.1)
    • Represent transformations in the plane using1 appropriate tools. (CCSS: G-CO.2)
    • Describe transformations as functions that take points in the plane as inputs and give other points as outputs. (CCSS: G-CO.2)
    • Compare transformations that preserve distance and angle to those that do not.2 (CCSS: G-CO.2)
    • Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS: G-CO.3)
    • Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS: G-CO.4)
    • Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools.3 (CCSS: G-CO.5)
    • Specify a sequence of transformations that will carry a given figure onto another. (CCSS: G-CO.5)
  2. Understand congruence in terms of rigid motions. (CCSS: G-CO)
    • Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. (CCSS: G-CO.6)
    • Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (CCSS: G-CO.6)
    • Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (CCSS: G-CO.7)
    • Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS: G-CO.8)
  3. Prove geometric theorems. (CCSS: G-CO)
    • Prove theorems about lines and angles.4 (CCSS: G-CO.9)
    • Prove theorems about triangles.5 (CCSS: G-CO.10)
    • Prove theorems about parallelograms.6 (CCSS: G-CO.11)
  4. Make geometric constructions. (CCSS: G-CO)
    • Make formal geometric constructions7 with a variety of tools and methods.8 (CCSS: G-CO.12)
    • Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (CCSS: G-CO.13)

Inquiry Questions:

  1. What happens to the coordinates of the vertices of shapes when different transformations are applied in the plane?
  2. How would the idea of congruency be used outside of mathematics?
  3. What does it mean for two things to be the same? Are there different degrees of “sameness?”
  4. What makes a good definition of a shape?

Relevance & Application:

  1. Comprehension of transformations aids with innovation and creation in the areas of computer graphics and animation.

Nature Of:

  1. Geometry involves the investigation of invariants. Geometers examine how some things stay the same while other parts change to analyze situations and solve problems.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians attend to precision. (MP)
  4. Mathematicians look for and make use of structure. (MP)

Content Area: Mathematics
Grade Level Expectations: Eighth Grade
Standard: 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. Transformations of objects can be used to define the concepts of congruence and similarity
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Verify experimentally the properties of rotations, reflections, and translations.1 (CCSS: 8.G.1)
  2. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.3)
  3. Demonstrate that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. (CCSS: 8.G.2)
  4. Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. (CCSS: 8.G.2)
  5. Demonstrate that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. (CCSS: 8.G.4)
  6. Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. (CCSS: 8.G.4)
  7. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.2 (CCSS: 8.G.5)

Inquiry Questions:

  1. What advantage, if any, is there to using the Cartesian coordinate system to analyze the properties of shapes?
  2. How can you physically verify that two lines are really parallel?

Relevance & Application:

  1. Dilations are used to enlarge or shrink pictures.
  2. Rigid motions can be used to make new patterns for clothing or architectural design.

Nature Of:

  1. Geometry involves the investigation of invariants. Geometers examine how some things stay the same while other parts change to analyze situations and solve problems.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians model with mathematics. (MP)

1 Lines are taken to lines, and line segments to line segments of the same length. (CCSS: 8.G.1a)
Angles are taken to angles of the same measure. (CCSS: 8.G.1b)
Parallel lines are taken to parallel lines. (CCSS: 8.G.1c)

2 For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. (CCSS: 8.G.5)

Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 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. Modeling geometric figures and relationships leads to informal spatial reasoning and proof
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Draw construct, and describe geometrical figures and describe the relationships between them. (CCSS: 7.G)
    • Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. (CCSS: 7.G.1)
    • Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. (CCSS: 7.G.2)
    • Construct triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. (CCSS: 7.G.2)
    • Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. (CCSS: 7.G.3)

Inquiry Questions:

  1. Is there a geometric figure for any given set of attributes?
  2. How does scale factor affect length, perimeter, angle measure, area and volume?
  3. How do you know when a proportional relationship exists?

Relevance & Application:

  1. The understanding of basic geometric relationships helps to use geometry to construct useful models of physical situations such as blueprints for construction, or maps for geography.
  2. Proportional reasoning is used extensively in geometry such as determining properties of similar figures, and comparing length, area, and volume of figures.

Nature Of:

  1. Mathematicians create visual representations of problems and ideas that reveal relationships and meaning.
  2. The relationship between geometric figures can be modeled
  3. Mathematicians look for relationships that can be described simply in mathematical language and applied to a myriad of situations. Proportions are a powerful mathematical tool because proportional relationships occur frequently in diverse settings.
  4. Mathematicians use appropriate tools strategically. (MP)
  5. Mathematicians attend to precision. (MP)

Content Area: Mathematics
Grade Level Expectations: Second Grade
Standard: 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 their attributes and used to represent part/whole relationships
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. (CCSS: 2.G.1)
  2. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. (CCSS: 2.G.1)
  3. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. (CCSS: 2.G.2)
  4. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. (CCSS: 2.G.3)
  5. Recognize that equal shares of identical wholes need not have the same shape. (CCSS: 2.G.3)

Inquiry Questions:

  1. How can we describe geometric figures?
  2. Is a half always the same size and shape?

Relevance & Application:

  1. Fairness in sharing depends on equal quantities, such as sharing a piece of cake, candy bar, or payment for a chore.
  2. Shapes are used to communicate how people view their environment.
  3. Geometry provides a system to describe, organize, and represent the world around us.

Nature Of:

  1. Geometers use shapes to describe and understand the world.
  2. Mathematicians reason abstractly and quantitatively. (MP)
  3. Mathematicians model with mathematics. (MP)