New Colorado P-12 Academic Standards

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Content Area: Mathematics
Grade Level Expectations: High School
Standard: 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:

2. Quantitative relationships in the real world can be modeled and solved using functions

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Construct and compare linear, quadratic, and exponential models and solve problems. (CCSS: F-LE)
    • Distinguish between situations that can be modeled with linear functions and with exponential functions. (CCSS: F-LE.1)
      • Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. (CCSS: F-LE.1a)
      • Identify situations in which one quantity changes at a constant rate per unit interval relative to another. (CCSS: F-LE.1b)
      • Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (CCSS: F-LE.1c)
    • Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.12 (CCSS: F-LE.2)
    • Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (CCSS: F-LE.3)
    • For exponential models, express as a logarithm the solution to \(ab^{ct} = d\) where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. (CCSS: F-LE.4)
  2. Interpret expressions for function in terms of the situation they model. (CCSS: F-LE)
    • Interpret the parameters in a linear or exponential function in terms of a context. (CCSS: F-LE.5)
  3. Model periodic phenomena with trigonometric functions. (CCSS: F-TF)
    • Choose the trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. * (CCSS: F-TF.5)
  4. Model personal financial situations
    • Analyze the impact of interest rates on a personal financial plan (PFL)
    • Evaluate the costs and benefits of credit (PFL)
    • Analyze various lending sources, services, and financial institutions (PFL)

Inquiry Questions:

  1. Why do we classify functions?
  2. What phenomena can be modeled with particular functions?
  3. Which financial applications can be modeled with exponential functions? Linear functions? (PFL)
  4. What elementary function or functions best represent a given scatter plot of two-variable data?
  5. How much would today’s purchase cost tomorrow? (PFL)

Relevance & Application:

  1. The understanding of the qualitative behavior of functions allows interpretation of the qualitative behavior of systems modeled by functions such as time-distance, population growth, decay, heat transfer, and temperature of the ocean versus depth.
  2. The knowledge of how functions model real-world phenomena allows exploration and improved understanding of complex systems such as how population growth may affect the environment , how interest rates or inflation affect a personal budget, how stopping distance is related to reaction time and velocity, and how volume and temperature of a gas are related.
  3. Biologists use polynomial curves to model the shapes of jaw bone fossils. They analyze the polynomials to find potential evolutionary relationships among the species.
  4. Physicists use basic linear and quadratic functions to model the motion of projectiles.

Nature Of:

  1. Mathematicians use their knowledge of functions to create accurate models of complex systems.
  2. Mathematicians use models to better understand systems and make predictions about future systemic behavior.
  3. Mathematicians reason abstractly and quantitatively. (MP)
  4. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  5. Mathematicians model with mathematics. (MP)

12 include reading these from a table. (CCSS: F-LE.2)

Content Area: Mathematics
Grade Level Expectations: Eighth Grade
Standard: 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:

3. Graphs, tables and equations can be used to distinguish between linear and nonlinear functions

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Define, evaluate, and compare functions. (CCSS: 8.F)
    • Define a function as a rule that assigns to each input exactly one output.5 (CCSS: 8.F.1)
    • Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (CCSS: 8.F.1)
    • Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).6 (CCSS: 8.F.2)
    • Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. (CCSS: 8.F.3)
    • Give examples of functions that are not linear.7
  2. Use functions to model relationships between quantities. (CCSS: 8.F)
    • Construct a function to model a linear relationship between two quantities. (CCSS: 8.F.4)
    • Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. (CCSS: 8.F.4)
    • Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.4)
    • Describe qualitatively the functional relationship between two quantities by analyzing a graph.8 (CCSS: 8.F.5)
    • Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.5)
    • Analyze how credit and debt impact personal financial goals (PFL)

Inquiry Questions:

  1. How can change best be represented mathematically?
  2. Why are patterns and relationships represented in multiple ways?
  3. What properties of a function make it a linear function?

Relevance & Application:

  1. Recognition that non-linear situations is a clue to non-constant growth over time helps to understand such concepts as compound interest rates, population growth, appreciations, and depreciation.
  2. Linear situations allow for describing and analyzing the situation mathematically such as using a line graph to represent the relationships of the circumference of circles based on diameters.

Nature Of:

  1. Mathematics involves multiple points of view.
  2. Mathematicians look at mathematical ideas arithmetically, geometrically, analytically, or through a combination of these approaches.
  3. Mathematicians look for and make use of structure. (MP)
  4. Mathematicians look for and express regularity in repeated reasoning. (MP)

5 Function notation is not required in 8th grade. (CCSS: 8.F.11)

6 For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. (CCSS: 8.F.2)

7 For example, the function \(A = s^2\) giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. (CCSS: 8.F.3)

8 e.g., where the function is increasing or decreasing, linear or nonlinear. (CCSS: 8.F.5)

Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 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:

2. Equations and expressions model quantitative relationships and phenomena

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form,2 using tools strategically. (CCSS: 7.EE.3)
  2. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies.3 (CCSS: 7.EE.3)
  3. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. (CCSS: 7.EE.4)
    • Fluently solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. (CCSS: 7.EE.4a)
    • Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.4 (CCSS: 7.EE.4a)
    • Solve word problems5 leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. (CCSS: 7.EE.4b)
    • Graph the solution set of the inequality and interpret it in the context of the problem. (CCSS: 7.EE.4b)

Inquiry Questions:

  1. Do algebraic properties work with numbers or just symbols? Why?
  2. Why are there different ways to solve equations?
  3. How are properties applied in other fields of study?
  4. Why might estimation be better than an exact answer?
  5. When might an estimate be the only possible answer?

Relevance & Application:

  1. Procedural fluency with algebraic methods allows use of linear equations and inequalities to solve problems in fields such as banking, engineering, and insurance. For example, it helps to calculate the total value of assets or find the acceleration of an object moving at a linearly increasing speed.
  2. Comprehension of the structure of equations allows one to use spreadsheets effectively to solve problems that matter such as showing how long it takes to pay off debt, or representing data collected from science experiments.
  3. Estimation with rational numbers enables quick and flexible decision-making in daily life. For example, determining how many batches of a recipe can be made with given ingredients, how many floor tiles to buy with given dimensions, the amount of carpeting needed for a room, or fencing required for a backyard.

Nature Of:

  1. Mathematicians model with mathematics. (MP)

2 whole numbers, fractions, and decimals. (CCSS: 7.EE.3)

3 For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. (CCSS: 7.EE.3)

4 For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? (CCSS: 7.EE.4a)

5 For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. (CCSS: 7.EE.4b)

Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 3. Data Analysis, Statistics, and Probability

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

Concepts and skills students master:

1. Statistics can be used to gain information about populations by examining samples

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Use random sampling to draw inferences about a population. (CCSS: 7.SP)
    • Explain that generalizations about a population from a sample are valid only if the sample is representative of that population.1 (CCSS: 7.SP.1)
    • Explain that random sampling tends to produce representative samples and support valid inferences. (CCSS: 7.SP.1)
    • Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. (CCSS: 7.SP.2)
    • Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. (CCSS: 7.SP.2)
  2. Draw informal comparative inferences about two populations. (CCSS: 7.SP)
    • Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.2 (CCSS: 7.SP.3)
    • Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.3 (CCSS: 7.SP.4)

Inquiry Questions:

  1. How might the sample for a survey affect the results of the survey?
  2. How do you distinguish between random and bias samples?
  3. How can you declare a winner in an election before counting all the ballots?

Relevance & Application:

  1. The ability to recognize how data can be biased or misrepresented allows critical evaluation of claims and avoids being misled. For example, data can be used to evaluate products that promise effectiveness or show strong opinions.
  2. Mathematical inferences allow us to make reliable predictions without accounting for every piece of data.

Nature Of:

  1. Mathematicians are informed consumers of information. They evaluate the quality of data before using it to make decisions.
  2. Mathematicians use appropriate tools strategically. (MP)

1 For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (CCSS: 7.SP.2)

2 For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. (CCSS: 7.SP.3)

3 For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. (CCSS: 7.SP.4)

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:

2. Concepts of similarity are foundational to geometry and its applications

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Understand similarity in terms of similarity transformations. (CCSS: G-SRT)
    • Verify experimentally the properties of dilations given by a center and a scale factor. (CCSS: G-SRT.1)
      • Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. (CCSS: G-SRT.1a)
      • Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (CCSS: G-SRT.1b)
    • Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. (CCSS: G-SRT.2)
    • Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (CCSS: G-SRT.2)
    • Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (CCSS: G-SRT.3)
  2. Prove theorems involving similarity. (CCSS: G-SRT)
    • Prove theorems about triangles.9 (CCSS: G-SRT.4)
    • Prove that all circles are similar. (CCSS: G-C.1)
    • Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS: G-SRT.5)
  3. Define trigonometric ratios and solve problems involving right triangles. (CCSS: G-SRT)
    • Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (CCSS: G-SRT.6)
    • Explain and use the relationship between the sine and cosine of complementary angles. (CCSS: G-SRT.7)
    • Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* (CCSS: G-SRT.8)
  4. Prove and apply trigonometric identities. (CCSS: F-TF)
    • Prove the Pythagorean identity \(sin^2(\theta) + cos^2(\theta) = 1\). (CCSS: F-TF.8)
    • Use the Pythagorean identity to find sin\((\theta)\), cos(\(\theta\)), or tan(\(\theta\)) given sin(\(\theta\)), cos(\(\theta\)), or tan(\(\theta\)) and the quadrant of the angle. (CCSS: F-TF.8)
  5. Understand and apply theorems about circles. (CCSS: G-C)
    • Identify and describe relationships among inscribed angles, radii, and chords.10 (CCSS: G-C.2)
    • Construct the inscribed and circumscribed circles of a triangle. (CCSS: G-C.3)
    • Prove properties of angles for a quadrilateral inscribed in a circle. (CCSS: G-C.3)
  6. Find arc lengths and areas of sectors of circles. (CCSS: G-C)
    • Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality. (CCSS: G-C.5)
    • Derive the formula for the area of a sector. (CCSS: G-C.5)

Inquiry Questions:

  1. How can you determine the measure of something that you cannot measure physically?
  2. How is a corner square made?
  3. How are mathematical triangles different from triangles in the physical world? How are they the same?
  4. Do perfect circles naturally occur in the physical world?

Relevance & Application:

  1. Analyzing geometric models helps one understand complex physical systems. For example, modeling Earth as a sphere allows us to calculate measures such as diameter, circumference, and surface area. We can also model the solar system, galaxies, molecules, atoms, and subatomic particles.

Nature Of:

  1. Geometry involves the generalization of ideas. Geometers seek to understand and describe what is true about all cases related to geometric phenomena.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians attend to precision. (MP)

9 Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (CCSS: G-SRT.4)

10 Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (CCSS: G-C.2)

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

Concepts and skills students master:

5. Objects in the real world can be modeled using geometric concepts

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Apply geometric concepts in modeling situations. (CCSS: G-MG)
    • Use geometric shapes, their measures, and their properties to describe objects.14 * (CCSS: G-MG.1)
    • Apply concepts of density based on area and volume in modeling situations.15 * (CCSS: G-MG.2)
    • Apply geometric methods to solve design problems.16 * (CCSS: G-MG.3)

Inquiry Questions:

  1. How are mathematical objects different from the physical objects they model?
  2. What makes a good geometric model of a physical object or situation?
  3. How are mathematical triangles different from built triangles in the physical world? How are they the same?

Relevance & Application:

  1. Geometry is used to create simplified models of complex physical systems. Analyzing the model helps to understand the system and is used for such applications as creating a floor plan for a house, or creating a schematic diagram for an electrical system.

Nature Of:

  1. 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.
  2. Mathematicians make sense of problems and persevere in solving them. (MP)
  3. Mathematicians reason abstractly and quantitatively. (MP)
  4. Mathematicians look for and make use of structure. (MP)

14 e.g., modeling a tree trunk or a human torso as a cylinder. (CCSS: G-MG.1)

15 e.g., persons per square mile, BTUs per cubic foot. (CCSS: G-MG.2)

16 e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. (CCSS: G-MG.3)

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:

2. Direct and indirect measurement can be used to describe and make comparisons

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Explain a proof of the Pythagorean Theorem and its converse. (CCSS: 8.G.6)
  2. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (CCSS: 8.G.7)
  3. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (CCSS: 8.G.8)
  4. State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (CCSS: 8.G.9)

Inquiry Questions:

  1. Why does the Pythagorean Theorem only apply to right triangles?
  2. How can the Pythagorean Theorem be used for indirect measurement?
  3. How are the distance formula and the Pythagorean theorem the same? Different?
  4. How are the volume formulas for cones, cylinders, prisms and pyramids interrelated?
  5. How is volume of an irregular figure measured?
  6. How can cubic units be used to measure volume for curved surfaces?

Relevance & Application:

  1. The understanding of indirect measurement strategies allows measurement of features in the immediate environment such as playground structures, flagpoles, and buildings.
  2. Knowledge of how to use right triangles and the Pythagorean Theorem enables design and construction of such structures as a properly pitched roof, handicap ramps to meet code, structurally stable bridges, and roads.
  3. The ability to find volume helps to answer important questions such as how to minimize waste by redesigning packaging or maximizing volume by using a circular base.

Nature Of:

  1. 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.
  2. Geometric objects are abstracted and simplified versions of physical objects
  3. Mathematicians make sense of problems and persevere in solving them. (MP)
  4. Mathematicians construct viable arguments and critique the reasoning of others. (MP)