## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties: solving problems by determining the lengths of the sides in right triangles; and finding the measurements of shapes by decomposing complex shapes into simpler ones. The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

## Standards

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Apply trigonometry to general triangles

Standard: (+) Prove the Laws of Sines and Cosines and use them to solve problems.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Apply trigonometry to general triangles

Standard: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Construct a tangent line from a point outside a given circle to the circle.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Identify and describe relationships among inscribed angles, radii, and chords. 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply theorems about circles

Standard: Prove that all circles are similar.

Degree of Alignment: Not Rated (0 users)

Cluster: Find arc lengths and areas of sectors of circles

Standard: 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; derive the formula for the area of a sector.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Define trigonometric ratios and solve problems involving right triangles

Standard: Explain and use the relationship between the sine and cosine of complementary angles.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Define trigonometric ratios and solve problems involving right triangles

Standard: Understand 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Prove theorems involving similarity

Standard: Prove theorems about triangles. 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Prove theorems involving similarity

Standard: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Define trigonometric ratios and solve problems involving right triangles

Standard: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: Verify experimentally the properties of dilations given by a center and a scale factor:

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Apply trigonometry to general triangles

Standard: (+) Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Find arc lengths and areas of sectors of circles

Indicator: 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; derive the formula for the area of a sector.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Construct a tangent line from a point outside a given circle to the circle.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Identify and describe relationships among inscribed angles, radii, and chords. 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Circles

Standard: Understand and apply theorems about circles

Indicator: Prove that all circles are similar.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: Verify experimentally the properties of dilations given by a center and a scale factor:

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Apply trigonometry to general triangles

Indicator: (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Apply trigonometry to general triangles

Indicator: (+) Prove the Laws of Sines and Cosines and use them to solve problems.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Apply trigonometry to general triangles

Indicator: (+) Derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Define trigonometric ratios and solve problems involving right triangles

Indicator: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Prove theorems involving similarity

Indicator: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Prove theorems involving similarity

Indicator: Prove theorems about triangles. 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Define trigonometric ratios and solve problems involving right triangles

Indicator: Understand 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Define trigonometric ratios and solve problems involving right triangles

Indicator: Explain and use the relationship between the sine and cosine of complementary angles.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and"Óif there is a flaw in an argument"Óexplain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

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# Tags (11)

- Mathematics
- Geometry and measures
- CCSS
- Circles
- Common Core Math
- Common Core PD
- Geometry
- ODE Learning
- Right Triangles
- Shapes
- Triangles

## Comments