Colorado Academic Standards Online

Colorado Essential Skills and Mathematical Practices:

1. Explore transformations in the plane using concrete and technological tools. The use of tools allows students to attend to precision. (MP5)
2. Use exact terms, symbols and notation when describing and working with geometric transformations. (MP6)

Inquiry Questions:

1. What is the relationship between functions and geometric transformations?
2. How is a figure’s symmetry connected to congruence transformations?

Coherence Connections:

1. This expectation supports the major work of high school.
2. In Grade 8, rotations, reflections, and translations are developed experimentally and students graph points in all four quadrants of the coordinate plane.
3. In high school, transformations are studied in terms of functions, where the inputs and outputs are points in the coordinate plane, and students understand the meaning of rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines, and line segments.
4. Geometric reasoning is expressed through formal proof and precise language, informal explanation and construction, and strategic experimentation to verify or refute claims.

Colorado Essential Skills and Mathematical Practices:

1. Justify claims of congruence in terms of rigid motions and follow others’ reasoning in describing alternate rigid motions that lead to the same congruence conclusion. (MP3)
2. Examine claims and make explicit use of definitions to support formal proof and justification of congruence relationships. (MP6)

Inquiry Questions:

1. How can transformations be used to prove that two triangles are congruent?
2. What is the minimum amount of information you need to know about two triangles in order to determine if they are congruent? Why is that the minimum?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 8, students study rigid motions using physical models or software, with an emphasis on geometric intuition, whereas high school geometry weighs precise definitions and geometric intuition equally.
3. In high school, students will compare graphs of functions and other curves to make congruence and similarity arguments based on rigid motions.

Colorado Essential Skills and Mathematical Practices:

1. Justify claims of congruence in terms of rigid motions, understand alternate reasoning, and recognize and address errors when appropriate. (MP3)
2. Make explicit use of definitions, symbols, and notation with lines, angles, triangles, and parallelograms. (MP6)

Inquiry Questions:

1. Can some theorems be proved without using other, previously proven theorems? If so, what does that imply about a system of theorems?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grades 7 and 8, students investigate properties of lines and angles, triangles, and parallelograms.
3. In high school, proof is sometimes formatted with a two-column approach, with one column headed “statements” and the other column headed “reasons.” Students may also write sentences (paragraph proof), or use boxes (flow proof), or they may employ other formats, or combine formats, for communicating proof.

Colorado Essential Skills and Mathematical Practices:

1. Use a variety of tools, appropriate to the task, to make geometric constructions. (MP5)
2. Precisely use construction tools and communicate their steps, reasoning, and results using mathematical language. (MP6)

Inquiry Questions:

1. How is a geometric construction like a proof?
2. How can you use properties of circles to ensure precise constructions?

Coherence Connections:

1. This expectation supports the major work of high school and includes an advanced (+) outcome.
2. In Grade 7, students draw geometric shapes with rulers, protractors, and technology.
3. In high school, students use proofs to justify validity of their constructions. They use geometric constructions to precisely locate the line of reflection between an image and its pre-image and to accurately draw a figure under a translation or rotation and justify its validity.

Colorado Essential Skills and Mathematical Practices:

1. Use auxiliary lines not part of the original figure when reasoning about similarity. (MP2)
2. Employ geometric tools and technology (including dynamic geometric software) in exploring and verifying the properties of dilations and in understanding the properties of similar figures. (MP5)
3. Connect algebraic and geometric content when using proportional reasoning to determine if two figures are similar. (MP7)
4. Recognize and use repeated reasoning in exploring and verifying the properties of dilations and similarity and in establishing the AA criterion for similar triangles. (MP8)

Inquiry Questions:

1. How can we use the concepts of similarity to measure real-world objects that are difficult or impossible to measure directly?
2. How are similarity and congruence related to one another?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 8, students informally investigate dilations and similarity, including the AA criterion.
3. In high school, students show that two figures are similar by finding a scaling transformation (dilation or composition of dilation with a rigid motion) or a sequence of scaling transformations that maps one figure to the other, and recognize that congruence is a special case of similarity where the scale factor is equal to \(1\).

Colorado Essential Skills and Mathematical Practices:

1. Justify reasoning using logical, cohesive steps when proving theorems and solving problems in geometry. (MP3)
2. Use precise geometric and other mathematical terms and symbols to construct proofs and solve problems in geometry. (MP6)
3. Maintain oversight of the problem-solving process and when writing proofs while attending to details and continually evaluating the reasonableness of intermediate results. (MP8)

Inquiry Questions:

1. How does the Pythagorean Theorem support the case for triangle similarity?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 7, students study proportional relationships and apply them to solve real-world problems.
3. In Grade 8, students are introduced to the concept of similar figures using physical models and geometry software.
4. In high school, students understand properties of similar triangles to develop understanding of right triangle trigonometry.

Colorado Essential Skills and Mathematical Practices:

1. Reason abstractly by relating the properties of similar triangles to the definitions of the trigonometric ratios for acute angles, recognizing that the proportionality of side measures creates a single ratio based on the angle measure, regardless of the size of the right triangle. (MP2)
2. Apply trigonometric ratios and the Pythagorean Theorem to model and solve real-world problems. (MP4)
3. Use structure to relate triangle similarity and the trigonometric ratios for acute angles. (MP7)

Inquiry Questions:

1. How are the trigonometric ratios for acute angles connected to the properties of similar triangles?
2. What visual representation(s) explains why the sine of an acute angle is equivalent to the cosine of its complement?

Coherence Connections:

1. This expectation represents major work of high school and includes a modeling (★) outcome.
2. In Grade 7, students apply proportional reasoning and solve problems involving scale drawings of geometric figures. In Grade 8, students connect proportional relationships to triangles representing the slope of a line and understand congruence and similarity using physical models, transparencies, or geometry software.
3. In high school, students apply their previous study of similarity to establish understanding of the trigonometric ratios for acute angles. They connect right triangle trigonometry to concepts with algebra and functions. They understand that trigonometric ratios are functions of the size of an angle, and use the Pythagorean Theorem to show that \(\sin^2(\theta) + \cos^2(\theta) = 1\).

Colorado Essential Skills and Mathematical Practices:

1. Make sense of the ambiguous case of the Law of Sines and persevere in determining valid and invalid solutions. (MP1)
2. Construct an argument proving the Laws of Sines and Cosines. (MP3)

Inquiry Questions:

1. Why does the formula \(A = \frac{1}{2}ab\sin(C)\) accurately calculate the area of a triangle?
2. In using the Law of Sines, when do we need to consider the ambiguous case?
3. How is the Law of Cosines related to the Pythagorean Theorem?

Coherence Connections:

1. This expectation represents advanced (+) work of high school.
2. In Grade 6, students learn to calculate the area of a triangle by developing the formula \(A = \frac{1}{2}bh\). In Grade 8, students understand and apply the Pythagorean Theorem.
3. In high school, students develop understanding of right triangle trigonometry through similarity. In advanced courses, students prove trigonometric identities using relationships between sine and cosine.

Colorado Essential Skills and Mathematical Practices:

1. Justify reasoning and use logical, cohesive steps when proving theorems and solving problems in geometry. (MP3)
2. Employ geometric tools and technology (including dynamic geometry software) in exploring relationships in circles and in circle-related constructions. (MP5)
3. Observe the relationships among angles in circles and extend their conclusions to a variety of scenarios. (MP7)

Inquiry Questions:

1. Draw or find examples of several different circles. In what ways are they related geometrically? How can you describe these relationships in terms of transformations?

Coherence Connections:

1. This expectation supports the major work of high school and includes an advanced (+) outcome.
2. In Grade 7, students informally derive and apply the equations of area and circumference of circles.
3. In high school, students relate circle properties to geometric constructions and proofs of their validity.

Colorado Essential Skills and Mathematical Practices:

1. Use verbal and written arguments using similarity to justify arc lengths and radian measures. (MP3)
2. Attend to precise mathematical definitions, relationships, and symbols to describe and solve problems involving arc lengths and areas of sectors of circles. (MP6)
3. Use understanding of the area of a circle and the meaning of a central angle to synthesize the formula for the area of a sector. (MP7)

Inquiry Questions:

1. In what ways is it more convenient to use radian measure for a central angle in a circle, rather than degree measure?

Coherence Connections:

1. This expectation represents advanced (+) work of high school.
2. In Grade 7, students know the formulas for area and circumference of a circle and apply proportional reasoning to real-world problems.
3. In high school, the formulas for area and circumference of a circle are generalized to fractional parts of a circle. Students apply proportional reasoning to find the length of an arc of a circle.

Colorado Essential Skills and Mathematical Practices:

1. Make conjectures about the form and meaning of an equation for a conic section, and plan a solution pathway that deliberately connects the geometric and algebraic representations of conic sections rather than simply jumping into a solution attempt. (MP1)
2. Use abstract and quantitative reasoning to apply the Pythagorean Theorem to the equations of conic sections, particularly circles and parabolas. (MP2)
3. Analyze the underlying structure of the equations for conic sections and their connection to the Pythagorean Theorem and to each other. (MP7)

Inquiry Questions:

1. How does the Pythagorean Theorem connect to the general equation for a circle with center \((a, b)\) and radius \(r\)? How can this be illustrated with a diagram?
2. How does the Pythagorean Theorem connect to the equation for a parabola? How can this be illustrated with a diagram?

Coherence Connections:

1. This expectation is in addition to the major work of high school and includes advanced (+) outcomes.
2. In Grade 8, students apply the Pythagorean theorem to find the length of an unknown side of a right triangle and calculate the distance between two points in the coordinate plane.
3. In high school, the application of the Pythagorean theorem is generalized to obtain formulas related to conic sections. Quadratic functions and completing the square are studied in the domain of interpreting functions. The methods are applied here to transform a quadratic equation representing a conic section into standard form.

Colorado Essential Skills and Mathematical Practices:

1. Connect coordinate proof to geometric theorems and the coordinate plane. (MP2)
2. Justify theorems involving distance and ratio, both verbally and written. (MP3)
3. Apply understandings of distance and perpendicularity to polygons. (MP7)

Inquiry Questions:

1. What mathematical concepts and tools become available when coordinates are applied to geometric figures?

Coherence Connections:

1. This expectation represents major work of high school and includes a modeling (★) outcome.
2. In Grade 8, students relate the slope triangles of a line to proportions and similarity, and they apply the Pythagorean theorem to determine distances in the coordinate plane.
3. In high school, students prove theorems using coordinates, and they use algebraic and geometric concepts to connect the equations of conic sections and their corresponding graphs.

Colorado Essential Skills and Mathematical Practices:

1. Formulate justifications of the formulas for circumference of a circle, area of a circle, and volumes of cylinders, pyramids, and cones. (MP3)
2. Apply volume formulas for cylinders, pyramids, cones, and spheres to real-world contexts to solve problems. (MP4)
3. Apply technologies, as appropriate, to estimate and compute areas and volumes. (MP5)

Inquiry Questions:

1. How could you use other geometric relationships to explain why the volume of a cylinder is \(V = \pi r^2 h\)?
2. How could you algebraically prove that a right cylinder and a corresponding oblique cylinder have the same volume?

Coherence Connections:

1. This expectation is in addition to the major work of high school and includes modeling (★) and advanced (+) outcomes.
2. In Grade 7, students informally derive the formula for the area of a circle from the circumference. In Grade 8, students know and use the formulas for volumes of cylinders, cones, and spheres.
3. In high school, students construct informal justifications of volume formulas. Students might view a pyramid as a stack of layers and, using Cavalieri’s Principle, see that shifting the layers does not change the volume. Furthermore, stretching the height of the pyramid by a given scale factor thickens each layer by the scale factor which multiplies its volume by that factor. Using such arguments, students can derive the formula for the volume of any pyramid with a square base.
4. Reasoning about area and volume geometrically prepares students for topics in calculus.

Colorado Essential Skills and Mathematical Practices:

1. Conceptualize problems using concrete objects or pictures, checking answers using different methods, and continually asking themselves, "Does this make sense?" (MP1)
2. Reason abstractly to visualize, describe, and justify their understanding of cross-sections of three-dimensional objects and of three-dimensional objects generated by rotations of two-dimensional objects without concrete representations. (MP2)

Inquiry Questions:

1. When will the shape of a cross-section of a three-dimensional object be the same for all planes that intersect the object? How do you know?

Coherence Connections:

1. This expectation supports the major work of high school.
2. In Grades 6–8, students apply geometric measurement to real-world and mathematical problems, making use of properties of figures as they dissect and rearrange them in order to calculate or estimate lengths, areas, and volumes.
3. In high school, students examine geometric measurement more closely, giving informal arguments to explain formulas, drawing on abilities developed in earlier grades: dissecting and rearranging two- and three-dimensional figures; and visualizing cross-sections of three-dimensional figures.
4. In calculus, students use integrals to calculate the volume of solids formed by rotating a curve around an axis.

Colorado Essential Skills and Mathematical Practices:

1. Make sense of real-world shapes and spaces by applying geometric concepts. (MP1)
2. Apply the properties and relationships associated with geometric figures and measurement to make sense of, reason about, and solve real-world problems. (MP4)
3. Model and solve problems involving geometric figures and measurement using technology and dynamic geometry software. (MP5)

Inquiry Questions:

1. What are all the ways you would use geometry to design a figure in 3D software, estimate the mass of printer filament needed to 3D print a \(\frac{1}{8}\)-scale model of your figure, then calculate the cost to produce the figure out of a different material at full size?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) outcomes.
2. In high school, geometric modeling can be used in Fermi problems, problems which ask for rough estimates of quantities and often involve estimates of densities.
3. In high school, students apply trigonometric measurement to many different authentic contexts. Of all the subjects students learn in geometry, trigonometry may have the greatest application in college and careers. Applying abstract geometric concepts involving congruence, similarity, measurement, trigonometry, and other related areas to solving problems situated in real-world contexts provides a means of building understanding about concepts and experiencing the usefulness of geometry.