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2 of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. 8.G.B Understand and apply the Pythagorean Theorem. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. DOK Levels: 1, 2 Achievement Level Descriptors: RANGE Level 1 Students should be able to draw or construct geometric Achievement Level shapes with given conditions by freehand, with ruler and protractor, Descriptor and by using technology. (Range ALD) Level 2 Students should be able to describe geometric shapes with Target E: given conditions, and determine whether or not a set of any three Draw, construct, and given angle or side-length measures can result in a unique triangle, describe geometrical more than one triangle, or no triangle at all. They should be able to figures, and describe describe the relationship between a geometric figure and its scale the relationships drawing by finding the scale factor between them. between them. Level 3 Students should be able to compute actual lengths and areas from a scale drawing, and reproduce a scale drawing using a different scale. They should be able to describe the two-dimensional figures that result from slicing prisms and pyramids by planes that are parallel to a given face. Level 4 Students should be able to describe the two-dimensional figures that result from slicing cones, spheres, cylinders, or other three-dimensional figures with rectangular or triangular faces by planes that are not parallel to a given face. 1. The student creates scale drawings. 2. The student solves problems involving scale drawings using proportional reasoning. 2 Version 2.0

4 Task Model 1 Graphing DOK Level 2 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 1. The student creates scale drawings. Tools: Calculator Prompt Features: The student is prompted to create a scale drawing of a polygon on a grid using drawing tools. Stimulus Guidelines: Item difficulty can be adjusted via these example methods: Figures may consist of polygons such as quadrilaterals, trapezoids or parallelograms. Lengths and angles may be positive integers or rational numbers. Scale factor may be a positive rational number. Inclusion of extraneous information. TM1a Stimulus: The student is presented with a simple polygon on a grid and a scale factor. Example Stem: A scale factor of 2 is applied to this figure. Use the Connect Line tool to draw the resulting figure. Interaction: The student is given the Connect Line, Add Point, and Delete tools to draw the polygon on a grid. Rubric: (1 point) Student draws the correct figure with correct dimensions. Allow for correct scoring regardless of orientation of the figure (see one example of a correct response below). Response Types: Graphing 4 Version 2.0

5 Task Model 1 Equation/Numeric DOK Level 2 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Prompt Features: The student is prompted to give the area of an actual figure based on a scale drawing and scale factor. Stimulus Guidelines: Item difficulty can be adjusted via these example methods: Types of polygons (square, rectangle, parallelogram, or right triangle). Linear dimensions can be a combination of rational numbers. Area can be a combination of rational numbers. TM1b Stimulus: The student is presented with a polygon (square, rectangle, parallelogram, or right triangle) on a grid and the scale factor at which it was created. Example Stem: The scale drawing of the right triangle shown was drawn using a scale factor of. 1. The student creates scale drawings. 2. The student solves problems involving scale drawings using proportional reasoning. Tools: Calculator Each square on the grid is 3 units in length. What is the area of the actual figure, in square units, on which this scale drawing is based? Rubric: (1 point) Student enters the correct area (e.g., 2700). Equation/Numeric 5 Version 2.0

6 Task Model 2 Equation/Numeric DOK Level 1 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 2. The student solves problems involving scale drawings using proportional reasoning. Tools: Calculator Prompt Features: The student is prompted to give the length of one or more sides of a polygon or the scale factor being applied based on a scale drawing. Stimulus Guidelines: Scale factor and side lengths may be positive rational numbers. Item difficulty can be adjusted via these example methods: o Types of polygons (square, rectangle, parallelogram, or right triangle). o Linear dimensions can be a combination of rational numbers. o Lengths of corresponding sides of similar polygons are not all labeled. o Inclusion of extraneous information. TM2a Stimulus: The student is presented with two polygons and a scale factor. A side length is given and the corresponding side is labeled with a variable. Example Stem: Figure A is a scale image of Figure B, as shown. The scale that maps Figure A onto Figure B is of x. 1 1 : 3. Enter the value 2 Rubric: (1 point) Student gives the correct value of the variable, which is a single numeric answer. Units, if given, should be assumed from the stem (e.g., 17.5). Equation/Numeric 6 Version 2.0

7 Task Model 2 Equation/Numeric DOK Level 1 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. TM2b Stimulus: The student is presented with two polygons with lengths of some or all corresponding sides given or indicated by a grid. Example Stem: Figure B is a scale image of Figure A, as shown. 2. The student solves problems involving scale drawings using proportional reasoning. Tools: Calculator Enter the scale factor applied to Figure A to produce Figure B. Rubric: (1 point) Student gives the correct scale factor, which is a single numeric answer. The keypad should only contain numbers (e.g., 3). Equation/Numeric 7 Version 2.0

8 Task Model 2 Equation/Numeric DOK Level 2 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Prompt Features: The student is prompted to give the length of one side of an actual figure based on a scale drawing. Stimulus Guidelines: Context should be familiar to year olds. Scale factor may be given in a key. Item difficulty can be adjusted via these example methods: o Numbers can be rational. o Combinations of area and length are provided. o Extra information is provided. o Unit conversion is used. o Scale factor may be a positive rational number. TM2c Stimulus: The student is presented with information about the area and/or dimensions of a scale drawing, including the scale factor. Example Stem: The front side of a playhouse is shown in this scale drawing. The height of the door in the drawing is 1.8 inches. 2. The student solves problems involving scale drawings using proportional reasoning. The scale that maps the drawing to the actual playhouse is 1 inch to 2.5 feet. Tools: Calculator Using the scale given, enter the actual height of the playhouse door, in feet. Rubric: (1 point) Correct answer is a single numeric answer (e.g., 4.5). Equation/Numeric 8 Version 2.0

9 Task Model 2 Equation/Numeric DOK Level 2 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Prompt Features: The student is prompted to give the area of a polygon based on a scale drawing. Stimulus Guidelines: Context should be familiar to year olds. Item difficulty can be adjusted via these example methods: o Dimensions can be a combination of positive rational numbers. o Scale factor may be a positive rational number. TM2d Stimulus: The student is presented with a scale drawing of a polygon (square, rectangle, or right triangle) with dimensions labeled and the dimension for one side of the actual polygon given. Example Stem: This scale drawing of a rectangular rug has dimensions 8 inches by 5 inches. The length of the longer side of the actual rug is 32 feet. 2. The student solves problems involving scale drawings using proportional reasoning. 8 in 5 in Tools: Calculator Enter the area, in square feet, of the actual rug. Rubric: (1 point) Correct answer is a single numeric answer. Units should be assumed from the stem (e.g., 640). Equation/Numeric ` 9 Version 2.0

10 Task Model 3 Matching Tables DOK Level 2 7.G.2 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. 3. The student draws, constructs, or describes geometric shapes given certain conditions. Tools: None Prompt Features: Identify the type of triangles that can be constructed based on given information about the sides and angles of the triangle. Stimulus Guidelines: Number of statements should be within three to five. Information given may either: o Describe a unique triangle o Be insufficient to determine a unique triangle (more than one triangle) o Be inconsistent with the triangle inequality or the triangle angle sum theorem (no triangle) Item difficulty can be adjusted via these example methods: o Triangles may be right, acute, obtuse, equilateral, isosceles, or scalene. o Angle measures and unit lengths are composed of differing positive rational numbers. o Inclusion of extraneous information. TM3a Stimulus: The student is presented with side lengths and/or angle measures of a triangle and statements about the type of triangle. Example Stem: A triangle has a 45 angle, a 60 angle, and a side 3 centimeters in length. Select True or False for each statement about this type of triangle. Statement True False The triangle might be an isosceles triangle. The triangle must be an acute triangle. The triangle must contain an angle measuring 75. Rubric: (1 point) Student selects True or False for each statement (e.g., F, T, T). Item Type: Matching Tables 10 Version 2.0

11 Task Model 3 Graphing DOK Level 1 7.G.2 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. Prompt Features: The student is prompted to generate geometric shapes based on given conditions. Stimulus Guidelines: Item difficulty can be adjusted via these example methods: Figures may consist of triangles (right, acute, obtuse, equilateral, isosceles, scalene), quadrilaterals, trapezoids or parallelograms, or combinations of the above. Combinations of given side lengths and angles. Inclusion of extraneous information. TM3b Stimulus: The student is presented with a series of conditions regarding a triangle or quadrilateral. The conditions should determine a unique polygon, and measurements should be positive integers reasonable for display in the workspace provided. Example Stem: Use the Connect Line tool to draw a triangle with a 90 angle, a side with a length of 7 units, and a side with a length of 4 units. Each square on the grid is 1 unit in length. Interaction: The student is given the Connect Line, Add Point, and Delete tools to generate line segments on a grid. 3. The student draws, constructs, or describes geometric shapes given certain conditions. Rubric: (1 point) The student correctly constructs the figure described. Tools: None Graphing 11 Version 2.0

12 Task Model 4 Multiple Choice, multiple correct response DOK Level 2 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. Prompt Feature: The student is prompted to identify a two-dimensional figure that results from slicing a given three-dimensional figure by a plane. Stimulus Guidelines: Item difficulty can be adjusted via these example methods: Three-dimensional figures can be right-rectangular prisms or right-rectangular pyramids. Slices may be horizontal planes, vertical planes, or planes that are not parallel to a face. Increasing number of possible answer choices. TM4 Stimulus: The student is presented with a three-dimensional figure and a description of how the figure is sliced by a plane. Example Stem: This figure is a square pyramid. 4. The student describes a two-dimensional figure resulting from slicing a three-dimensional figure by a plane. Tools: None Select all figures that can be formed by a vertical slice perpendicular to the base of the square pyramid. A. Isosceles Trapezoid B. Line segment C. Square D. Triangle Answer Choices: Answer choices will be names of polygons and can also include line segment as a choice. Rubric: (1 point) Student selects the correct figures (e.g., A, B, and D). Multiple Choice, multiple correct response 12 Version 2.0

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