Grade 7UNIT 6:Geometrysuggested Number of Days for Entire UNIT: 35

Grade 7UNIT 6:GeometrySuggested Number of Days for Entire UNIT: 35

Essential Questions / Key Concepts / Cross Curricular Connections
How do angle relationships apply to 2-dimensional and 3-dimensional figures?
How are circumference, diameter, and pi related?
Correspondence
Unique Triangle
Identical Triangles
Three sides condition
Two angles and the included side condition
Two angles and the side opposite a given angle condition
Two sides and the included angle condition
Two sides and a non-included angle condition
Right rectangular pyramid
Surface of a pyramid
Supplementary
Scale models
Vertical
Complementary
Adjacent
Plane section /

Unknown Angles
Constructing Triangles
Slicing Solids
Problems Involving Area and Surface Area
Problems Involving Volume

*Assessment and Review
Mid-Module Assessment and Review: After Section B (4 days, included in Unit Instructional Days)
End-of-Module Assessment and Review: After Section E (4 days, included in Unit Instructional Days) / Social Studies: Have students investigate and determine the area of various territories/acquisitions that occurred throughout the growth and expansion of the United States. Various maps and their scales will be used to determine the given areas. Throughout the year the areas will be compared in size order.
Science: When learning about the human body, have students look at various body systems and research the circumference of the skull at various ages and its relationship with growth and development. Also explore abnormalities associated with atypical measurements.
Unit Outcome (Focus)
Students delve further into several geometry topics they have been developing over the years. Grade 7 presents some of these topics (e.g., angles, area, surface area, and volume) in the most challenging form students have experienced yet. Unit 6 assumes students understand the basics; the goal is to build a fluency in these difficult problems. The remaining topics (i.e., working on constructing triangles and taking slices (or cross sections) of three-dimensional figures) are new to students.

Grade 7 UNIT 6SECTION A: Unknown AnglesSuggested Number of Days for SECTION: 4

Essential Question / Key Concept / Standards for Mathematical Practice
How can you use facts about angles to solve simple equations? /

Complementary and Supplementary Angles
Solve for Unknown Angles using Equations

/ 1. Make sense of problems and persevere in solving them
6. Attend to precision
7. Look for and make use of structure
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Students solve for unknown angles. The most challenging examples of unknown angle problems (both diagram-based and verbal) require students to use a synthesis of angle relationships and algebra. The problems are multi-step, requiring students to identify several layers of angle relationships and to fit them with an appropriate equation to solve. Unknown angle problems show students how to look for, and make use of, structure (MP.7). In this case, they use angle relationships to find the measurement of an angle. / 7.G.5
(DOK 2) / Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. / 

Archdiocese of New YorkPage 12014-2015

Grade 7 UNIT 6 SECTION B: Constructing TrianglesSuggested Number of Days for SECTION: 11

Essential Question / Key Concepts / Standards for Mathematical Practice
How can you construct geometric shapes? /

Unique Triangles
Drawing Geometric Shapes
Drawing Parallelograms
Drawing Triangles
Conditions for a Unique Triangle―Three Sides and Two Sides and the Included Angle
Conditions for a Unique Triangle—Two Angles and a Given Side Conditions on Measurements that Determine a Triangle
Unique Triangles―Two Sides and a Non-Included Angle Checking for Identical Triangles
Using Unique Triangles to Solve Real-World and Mathematical Problems

/ 3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
7. Look for and make use of structure
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Students work extensively with a ruler, compass, and protractor to construct geometric shapes, mainly triangles (7.G.A.2). The use of a compass is new (e.g., how to hold it, and to how to create equal segment lengths). Students use the tools to build triangles, provided given conditions, such side length and the measurement of the included angle (MP.5). Students also explore how changes in arrangement and measurement affect a triangle, culminating in a list of conditions that determine a unique triangle. Students notice the conditions that determine a unique triangle, more than one triangle, or no triangle. / 7.G.2
(DOK 3) / Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
. / 

Grade 7 UNIT 6SECTION C: Slicing SolidsSuggested Number of Days for SECTION: 4

Essential Question / Key Concept / Standards for Mathematical Practice
How can you describe two-dimensional figures that result from slicing three-dimensional figures? /

Slicing a Right Rectangular Prism with a Plane
Slicing a Right Rectangular Pyramid with a Plane
Slicing on an Angle
Understanding Three-Dimensional Figures

/ 1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Students begin exploring cross sections, or slices, of three-dimensional shapes and examining the two-dimensional results of different kinds of slices. Students learn what it means to slice a three-dimensional figure with a plane and examine slices made parallel to the base of right rectangular prisms and pyramids. Students slice the prisms and pyramids with vertical slices so that the plane meets the base in a line segment. Students experiment with skewed slices and try to predict how to slice figures to yield particular shapes, like how to slice a cube in order to get a cross section shaped like a triangle or a pentagon. Students learn that the slices can be used to determine the number of cubes in the figure. / 7.G.3 / Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. / 

Archdiocese of New YorkPage 12014-2015

Grade 7 UNIT: 6 SECTION: D: Problems Involving Area and Surface Area Suggested Number of Days for SECTION: 5

Essential Question / Key Concept / Standards for Mathematical Practice
How can you solve real-world problems involving area, volume and surface area of two and three-dimensional objects? /

Real-World Area Problems
Mathematical Area Problems
Area Problems with Circular Regions
Surface Area

/ 4. Model with mathematics
5. Use appropriate tools strategically
7. Look for and make use of structure
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Students use area problems embedded in real-world context, for example, finding the cost of carpeting a home based on a floor plan, calculating the cost of seeding a lawn, and determining how many stop signs can be painted with a given amount of paint. Students use the area properties to justify the repeated use of the distributive property. Students apply their knowledge of finding the area of both polygons and circles to find the area of composite figures, some of which have “holes” or missing sections in the form of geometric shapes. Students find the surface area of basic and composite three-dimensional figures. / 7.G.6
(DOK 2) / Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. / 

POSSIBLE ACTIVITIES/RESOURCES

SCALING AND DRAWING: Provide students with a scaled image of house or room. Challenge students to reproduce the image using a scale factor of 3. This exercise can be reversed using a large blueprint and using a scale factor of 5.

SCALING CUBES/ A SENSE OF SCALE: One of the basic ideas about scale is how changing the length, width, and heightof a three-dimensional object affects its surface area and its volume. In this exercise,students will build bigger and bigger cubes to understand these scaling relationships.

SHRINKING PROBLEM: When working on a report for class, Catrina read that a woman over the age of 40 can lose approximately 0.06 centimeters of height per year.

Catrina's Aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54 centimeters.)
Catrina's 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother's height has also decreased at this rate, about how tall was she at age 40? Explain your reasoning.

CUBES: Fill a box with cubes, rows of cubes, or layers of cubes. The number of unit cubes needed to fill the entire box is known as the volume of the box. Can you determine a rule for finding the volume of a box if you know its width, depth, and height?

IS IT POSSIBLE? Instruct students to draw particular shapes with given clues and parameters. If it is not possible to draw a shape with the givenparameters have them explain why. Demonstrate with examples and have students write one example about triangles.

Ex: Draw a quadrilateral with only one set of parallel lines and right angles. Is this possible? Draw a quadrilateral with two sets of parallel lines andone right angle. Draw a quadrilateral with one set of parallel lines and no right angles.

FRACTALS: Challenge students to research fractals and describe and/or create one of the following fractals, or their own fractal: Sierpinski triangle,

Koch edge, Peano curve, Lorenz attractor, Dragon curve, etc.

STATUE OF LIBERTY ACTIVITY: Have the students measure the length of their arm and their nose, using the same unit or measure. Assume that the Statue of Liberty is similar in scale to an average person. In this case, the lengths of corresponding body parts should have the same ratio. To estimate the lengths for an average person, average the class measurements. Answers will vary depending on measurement errors and what was used as an average. Have students then compare the ratio of their arm and nose to that of the Statue of Liberty. Is the ratio the same? Next compare the average ratio. Was that more accurate? Why or why not? Information about the Statue of Liberty can be found online:

SAND UNDER THE SWING SET: The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

FINDING SURFACE AREA AND VOLUME: Using the isometric drawing tool, students build three-dimensional figures and find the surface area and volume of each figure. Before using the isometric drawing tool, it would be helpful to review volume and surface area of three dimensional figures, namely ones built from cubes. In pairs, students should discuss their own definitions and give examples of the uses of each. Students can share their definitions and examples with the class:

ISOMETRIC DRAWING TOOL: Use this interactive tool to create dynamic drawings on isometric dot paper. Draw figures using edges, faces, or cubes. You can shift, rotate, color, decompose, and view in2D or3D. Start by clicking on the cube along the left side; then, place cubes on the grid where you would like them:

SQUARE CIRCLES: An online lesson that allows students to use a variety of units when measuring the side length and perimeter of squares, and the diameter and circumference of circles. From these measurements, students will discover the constant ratio of1:4 for all squares and the ratio of approximately1:3.14 for all circles.Lessons can be found online:

PERFORMANCE TASKS:

PROBLEM OF THE WEEK: A weekly challenge problem is located at Click on For Students to find challenge.

APPS:

ITOOCH JUNIOR HIGHSCHOOL, byeduPAD, has more than 10,000 exercises in Math (properties and operations, graphs, algebra, geometry, statistics and probability, and data analysis).

Archdiocese of New YorkPage 12014-2015