## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to visualize two-dimensional cross-sections of representations of three-dimensional objects. In particular, the lesson will help you identify and help students who have difficulties recognizing and drawing two-dimensional cross-sections at different points along a plane of a representation of a three-dimensional object.

- 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

# Comments

## Standards

Cluster: Solve real-world and mathematical problems involving area, surface area, and volume

Standard: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Degree of Alignment: 3 Superior (1 user)

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: 2 Strong (1 user)

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: 2 Strong (1 user)

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: 2 Strong (1 user)

Cluster: Mathematical practices

Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: 2 Strong (1 user)

# 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: 2 Strong (1 user)

# 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: 2 Strong (1 user)

# 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: 2 Strong (1 user)

# Common Core State Standards Math

Grade 6,Ratios and Proportional RelationshipsCluster: Understand ratio concepts and use ratio reasoning to solve problems

Standard: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Degree of Alignment: 2 Strong (1 user)

# 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: 2 Strong (1 user)

Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions

Standard: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Degree of Alignment: 2 Strong (1 user)

Learning Domain: Expressions and Equations

Standard: Apply and extend previous understandings of arithmetic to algebraic expressions

Indicator: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Degree of Alignment: 2 Strong (1 user)

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: 2 Strong (1 user)

Learning Domain: Ratios and Proportional Relationships

Standard: Understand ratio concepts and use ratio reasoning to solve problems

Indicator: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Degree of Alignment: 2 Strong (1 user)

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: 2 Strong (1 user)

Learning Domain: Geometry

Standard: Solve real-world and mathematical problems involving area, surface area, and volume

Indicator: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Degree of Alignment: 2 Strong (1 user)

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: 2 Strong (1 user)

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: 2 Strong (1 user)

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: 2 Strong (1 user)

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: 2 Strong (1 user)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: 2 Strong (1 user)

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: 2 Strong (1 user)

Learning Domain: Ratios and Proportional Relationships

Standard: Understand ratio concepts and use ratio reasoning to solve problems.

Indicator: Use ratio and rate reasoning to solve real-world and mathematical problems.

Degree of Alignment: 2 Strong (1 user)

## Evaluations

# Achieve OER

Average Score (3 Points Possible)Degree of Alignment | 2.0 (1 user) |

Quality of Explanation of the Subject Matter | 3 (1 user) |

Utility of Materials Designed to Support Teaching | 3 (1 user) |

Quality of Assessments | 3 (1 user) |

Quality of Technological Interactivity | N/A |

Quality of Instructional and Practice Exercises | 2 (1 user) |

Opportunities for Deeper Learning | 3 (1 user) |

# EQuIP Rubric

Average Score (3 Points Possible)ELA | Math |

Alignment to the Rigor of the CCSS | 3 (1 user) |

Key Shifts in the CCSS | 3 (1 user) |

Instructional Supports | 2 (1 user) |

Assessment | 2 (1 user) |

Overall Rating for the Lesson/Unit | E/I (1 user) |

Alignment to the Rigor of the CCSS | N/A |

Key Shifts in the CCSS | N/A |

Instructional Supports | N/A |

Assessment | N/A |

Overall Rating for the Lesson/Unit | N |

I like the inclusion of all students. I would like to have seen some technology included but the real life examples exemplified the lesson and related it to the students. I would also like to know how much time you spent on each section and any differentiated modifications you made. Overall, a great group activity!

Sharing Costs: Travelling to School - Students solve a problem using real-world modeling involving proportional relationships. Lesson plans include instructional strategies and sample questions designed to increase student conceptional understanding.

Laws of Arithmetic: Task provides teachers and students with assessment data to determine how students perform arithmetic operations by recognizing and applying the order of operations. Students apply distributive and commutative properties, write and evaluate numerical expressions.

2D Representations of 3D Objects: Unit asks students to explore relationships between two- and three-dimensional objects by identifying associated shapes of two-dimensional cross sections of three-dimensional objects (i.e., the shape of water level in given objects.

Lesson plans provide instructional strategies along with sample questioning strategies to build student understanding. It also provides potential student misconceptions and questions designed to build understanding.

Authors of the materials suggest alignment to G-GMD 1, G-GMD 2, and G-GMD 3. However, the materials do not address volume formulas, nor do they address Cavalieri’s principle.

The unit demonstrates a strong alignment to G-GMD 4.

Instructional strategies build opportunities for students to utilize mathematical practices 3, 6, and 7. Mathematical Practice 5 was rated as a level 1 because the current lesson design does not require students to determine appropriate / strategic use of available tools for the learning process (i.e., students are given tools to use).