## Description

- Overview:
- This lesson unit is intended to help teahcers assess how well students solve problems involving measurement, and in particular, to identify and help students who have the following difficulties; computing measurements using formulas; decomposing compound shapes into simpler ones; using right triangles and their properties to solve real-world problems.

- 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: 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: 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: 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)

# 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)

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: 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: Geometric Measurement and DimensionCluster: Explain volume formulas and use them to solve problems

Standard: (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Geometric Measurement and DimensionCluster: Explain volume formulas and use them to solve problems

Standard: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Geometric Measurement and DimensionCluster: Explain volume formulas and use them to solve problems

Standard: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

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: 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)

# Common Core State Standards Math

Grades 9-12,Geometry: Geometric Measurement and DimensionCluster: Visualize relationships between two-dimensional and three-dimensional objects

Standard: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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: 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)

Learning Domain: Geometry: Geometric Measurement and Dimension

Standard: Visualize relationships between two-dimensional and three-dimensional objects

Indicator: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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: (+) 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: 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: 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: Geometric Measurement and Dimension

Standard: Explain volume formulas and use them to solve problems

Indicator: Use volume formulas for cylinders, pyramids, cones, and spheres 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: (+) 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: Geometric Measurement and Dimension

Standard: Explain volume formulas and use them to solve problems

Indicator: (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

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: Geometric Measurement and Dimension

Standard: Explain volume formulas and use them to solve problems

Indicator: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit 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)

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)

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# Tags (11)

- Mathematics
- Geometry and measures
- CCSS
- Common Core Math
- Common Core PD
- Data and Measurement
- Geometry
- ODE Learning
- Real World Math
- Right Triangles
- Math Literacy Lessons

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