Mathematics Diagnostic Geometry Scoring Guide
Mathematics Geometry Mathematics, Diagnostic Geometry In participating districts, all students in grades 3 8, and high school Algebra I and Geometry, will take the LEAP 360 mathematics diagnostic assessments, which are designed to: identify the specific prerequisite skills individual students or groups of students need in order to be successful with major content for the current grade; help teachers to understand student performance on previous grade-level content that is prerequisite knowledge for the current grade; and assist teachers with meaningful, yet ambitious, goal setting for student learning targets. The purpose of this Scoring Guide is to provide teachers with the necessary information, guidance, and tools to score and interpret students responses to Reasoning (Type II) and Modeling (Type III) Constructed-Response (CR) items that align to Louisiana Student Mathematics Standards. The CRs, scoring rubrics, and numerous samples of student responses have been selected to ensure that teachers score actual responses fairly, accurately, and consistently. This document provides the scoring information and practice scoring exercise for the two CRs in the Geometry Diagnostic Mathematics assessment: Item 37: Modeling Item 55: Reasoning There are 8 anchor papers selected to illustrate the types of student responses that earn each possible number of points, or score, for each item. Each anchor paper is annotated to describe the rationale for the earned score. Scorers should: Review the alignment of the item (Evidence Statement and Standard[s]) as well as the metadata (Point Value, Depth of Knowledge [DOK], and Difficulty). Review the item. Review the rubric. Read each bullet point and each score point descriptor carefully. Read the student work and annotated scoring notes for each anchor paper. 1
Mathematics Geometry, Item 37 Alignment Mathematics, Diagnostic Geometry Task Type: Modeling (Type III) Evidence Statement: LEAP.III.GM.1: Solve multi-step contextual problems with degree of difficulty appropriate to the course, requiring application of knowledge and skills articulated in 6.G, 7.G, and/or 8.G. Primary Standard: 8.G.C.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Secondary Standard: 8.G.B.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Point Value: 3 DOK: 2 Difficulty: Medium 2
Mathematics Geometry, Item 37 Constructed-Response Item An artist has a block of clay in the shape of a cube. The edges of the cube measure 3 inches. The artist will use the clay to make models of pine trees. Each tree will be a solid cone with a base diameter of 1.5 inches and a height of 2 inches. Part A Determine the greatest number of clay pine trees that the artist can make. Show your work. Part B The artist plans to decorate each clay pine tree with pieces of yarn that will extend directly from the top of the tree to the base of the tree. Write an equation that can be used to determine the length, x, in inches, of each piece of yarn that the artist will need. Explain your equation. 3
Mathematics Geometry, Item 37 Scoring Information Mathematics, Diagnostic Geometry Part A (2 points) Correct answer (1 point) Valid work shown (1 point) 22 (clay pine trees) Sample Student Response: Volume of cube of clay: 27 in. 3, because 3 3 3 = 27 Volume of clay pine tree: 0.375π in. 3, because 1 2 ( 0.75) ( 2) 0.375 3 π = π Number of clay pine trees: 22, because 27 0.375π 22.9 Part B (1 point) Valid equation and explanation (1 point) Sample Student Response: 2 2 x = 2 + 0.75 ( ) 2 The variable x represents the slant height of the cone-shaped clay tree. This equation uses the radius of the base of the cone and the height of the cone, along with the Pythagorean Theorem, and can be used to determine the value of x. OR equivalent 3 The student earns 3 points. 2 The student earns 2 points. 1 The student earns 1 point. 0 The student s response is incorrect, irrelevant to the skill or concept being measured, or blank. 4
Mathematics Geometry, Item 37 Anchor Set Mathematics, Diagnostic Geometry The sample Geometry, Item 37, student responses or anchor set included in this section of the Scoring Guide are provided to ensure that teachers understand how to apply the rubrics reliably and consistently. The anchor set includes annotated references to both the rubric and specific examples from the student responses to exemplify why the response received a particular score. 5
Anchor Paper #1 Part A: 3 3 3 27 2 3.14 0.75 2 3.5325 3.5325 1.1775 3 27 about 22.9 1.1775 They can make 22 trees. Part B: 2 2 0.75 2 x 2 because of the Pythagorean theorem, but I used x instead of c. Score Information: 3 The response to Part A includes the correct answer (1) and valid work shown (1). The response to Part B includes a valid equation and explanation (1). 6
Anchor Paper #2 A: 22 cones 3 3 3 27 2 0.75 2 3.5325 3.5325 1.1775 3 27 22.9, but that means I can only make 22 whole cones. 1.1775 2 2 2 B: x 2 0.75 0.75 is the base of the triangle because it s the radius of the cone, and 2 is the height of the triangle because that s how tall the cone is. Score Information: 3 The response to Part A includes the correct answer (1) and valid work shown (1). The response to Part B includes a valid equation and explanation (1). 7
Anchor Paper #3 A. 1.5 2 1 2 0.75 2 1.18 3 3 3 3 27 27 1.18 22.9 The greatest number of clay pine trees the artist can make is 22. B. 2 x is the equation because the height of each tree is 2 inches, so each piece would also be 2 inches. Score Information: 2 The response to Part A includes the correct answer (1) and valid work shown (1). The response to Part B includes an incorrect equation and explanation (0). 8
Anchor Paper #4 A. The artist can make 5 clay pine trees. 3 3 3 27 2 1.5 2.25 2.25 2.355 3 2.355 2 4.71 27 5.7 5 4.71 B. 2 2 2 1.5 2 x To find x (the hypotenuse) do the pythagorean theorem Score Information: 2 The response to Part A includes an incorrect answer due to use of diameter instead of radius (0), but valid process shown (1). The response to Part B includes a valid equation (consistent with the error made in Part A) and explanation (1). 9
Anchor Paper #5 1 A. V 3 2 rh 3 2, because 2, and the diameter of the base of the cone will fit 1.5 twice across the edge of the cube B. The artist would need to use the equation 2 2 2 2 0.75 x because the height of the tree is 2 in and the radius of the base is 0.75 in. Score Information: 1 The response to Part A includes an incorrect answer (0) and incorrect work shown (0). The response to Part B includes a valid equation and explanation (1). 10
Anchor Paper #6 Part A: The greatest # of clay Pine trees the artist can make is 5. 1 2 1.5 2 4.712 3 3 3 3 27 27 5.73 4.712 Part B: 2x 1.5 x x is the # of yarn they have. Score Information: 1 The response to Part A includes an incorrect answer due to use of diameter instead of radius (0), but valid process shown (1). The response to Part B includes an incorrect equation and explanation (0). 11
Anchor Paper #7 1 0.75 2 h this is the cone formula. 3 1.17 Score Information: 0 The response to Part A includes no correct answer (0) and incomplete work shown (0). The response to Part B includes no correct equation and no explanation (0). 12
Anchor Paper #8 Part A = 3 3 9 9 6 1.5 You can fit 6 clay pine trees 9 Part B = x 6y Score Information: 0 The response to Part A includes an incorrect answer (0) and incorrect work shown (0). The response to Part B includes an incorrect equation and no explanation (0). 13
Mathematics Geometry, Item 37 Practice Scoring Exercise Five (5) sample responses have been selected and presented here to help scorers calibrate their expectations and judgments and to ensure student responses are accurately and consistently scored. Scorers should: Review the rubric again. Read each bullet point and each score point descriptor carefully. Read each sample response. Give each sample response a score based on the rubric. Compare your scores with the key, noting any differences in how the responses were scored. Begin scoring student responses when confident that the rubric can be applied accurately and consistently. 14
Mathematics Geometry, Item 37, Practice Scoring Exercise Paper Score Justification for Score #1 #2 #3 #4 #5 15
Practice Paper #1 A- 22 trees 27 1.177 B- I need the outside part of the cone, so I can use the Pythagorean therem, but use x instead of c. 2 2 0.75 2 x 2, x is about 2.1 inches. 16
Practice Paper #2 2 trees I divided the volume of the cube by the volume of the cone. 1 2 V 3.14 0.75 2 3 V 1.775 3 2.54 1.775 17
Practice Paper #3 Part A- 22 I found the volume of the cube. 3 3 3 27. Then I found the 2 3.14 0.75 2 volume of one cone. 1.1775. Then I divided 3 27 1.1775. Part B- 2 2 2 2 0.75 x because 2 is the height of a triangle, 0.75 is the length of the base, and x is the slant height. 18
Practice Paper #4 I need to find the distance from the top of the tree to the bottom 2 2 2 2 2 2 edge, so I used a b c and got x 2 0.75, so x 2.1. 19
Practice Paper #5 2 A. 3 6 54 1 3.14 2.75 2 1.1775 3 45 trees B. x 54 1.1775 20
Mathematics Geometry, Item 37 Practice Scoring Exercise Key Paper Score Justification for Score #1 2 The response to Part A includes the correct answer (1) and incomplete work shown (0). The response to Part B includes a valid equation and explanation (1). #2 1 The response to Part A includes an incorrect answer due to using 3 as the volume of the cube (0), but valid process shown (1). The response to Part B includes no equation and no explanation (0). #3 3 The response to Part A includes the correct answer (1) and valid work shown (1). The response to Part B includes a valid equation and explanation (1). #4 1 The response to Part A includes no correct answer (0) and no work shown (0). The response to Part B includes a valid equation and explanation (1). #5 0 The response to Part A includes an incorrect answer (0) and incomplete work shown (0). The response to Part B includes an incorrect equation and no explanation (0). 21
Mathematics Geometry, Item 55 Alignment Mathematics, Diagnostic Geometry Task Type: Reasoning (Type II) Evidence Statement: LEAP.II.8.5: Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content Scope: Knowledge and skills articulated in 8.G.2, 8.G.4 Primary Standard: 8.G.A.2: Explain 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. (Rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8.) Secondary Standard: 8.G.A.4: Explain 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. (Rotations are only about the origin, dilations only use the origin as the center of dilation, and reflections are only over the y-axis and x-axis in Grade 8.) Point Value: 3 DOK: 2 Difficulty: Medium 22
Mathematics Geometry, Item 55 Constructed-Response Item Two trapezoids are shown on this coordinate plane. Part A Identify a sequence of transformations that will show that trapezoid 1 must be congruent to trapezoid 2. Explain why the trapezoids must be congruent. Part B A dilation, with center of dilation at the origin, will be performed on trapezoid 1 to create trapezoid 3, which is not shown. Explain why trapezoid 3 must be similar to trapezoid 1, even though the scale factor is unknown. 23
Mathematics Geometry, Item 55 Scoring Information Mathematics, Diagnostic Geometry Part A (2 points) Correct sequence of transformations (1 point) Valid explanation of congruence (1 point) Sample Student Response: Two transformations will map trapezoid 1 onto trapezoid 2: a reflection across the y-axis, followed by a translation 8 units down. The trapezoids are congruent because reflections and translations are transformations in which the side lengths and the angle measures of a figure are preserved. Part B (1 point) Valid explanation of why trapezoids 1 and 3 must be similar (1 point) Sample Student Response: A dilation is a transformation in which the angle measures of a figure are preserved. A dilation changes the size of a figure if the scale factor is other than 1, but keeps the corresponding sides proportional. Having congruent angles and proportional sides will make the trapezoids similar, no matter what the scale factor of the dilation is. 3 The student earns 3 points. 2 The student earns 2 points. 1 The student earns 1 point. 0 The student s response is incorrect, irrelevant to the skill or concept being measured, or blank. 24
Mathematics Geometry, Item 55 Anchor Set Mathematics, Diagnostic Geometry The sample Geometry, Item 55, student responses or anchor set included in this section of the Scoring Guide are provided to ensure that teachers understand how to apply the rubrics reliably and consistently. The anchor set includes annotated references to both the rubric and specific examples from the student responses to exemplify why the response received a particular score. 25
Anchor Paper #1 a) The trapezoids must be congruent b/c they have the same distances between their points. sequence of transformations for 1 to 2: 1. translate 8 units down 2. reflection across the y-axis b) The scale factor will be the same for all sides, making the two figures (1 & 3) proportional, making the 2 figures similar Score Information: 3 The response to Part A includes a correct sequence of transformations (1) and a valid explanation of congruence (1). The response to Part B includes a valid explanation of similarity (1). 26
Anchor Paper #2 A- reflect over y-axis, then translate down 8 units Reflections and translating always keep the shape the same size and shape. B- Dilating makes a shape larger or smaller or the same size but always proportional. Score Information: 3 The response to Part A includes a correct sequence of transformations (1) and a valid explanation of congruence (1). The response to Part B includes a valid explanation of similarity (1). 27
Anchor Paper #3 A.) Reflection across the y-axis, then slide 8 units down. They are congruent because they didn t dilate it. B.) For a dilation the size changes, but the sides are always proportional. Score Information: 2 The response to Part A includes a correct sequence of transformations (1), but an incomplete explanation of congruence (0). The response to Part B includes a valid explanation of similarity (1). 28
Anchor Paper #4 Part A. Trapezoid 1 must be congruent to trapezoid 2 because you can move trapezoid 2 onto Trapezoid 1 and its the exact same size and shape. Part B. Trapezoid 3 must be similar to Trapezoid 1 because dilated means making the sides bigger or smaller but keeping the angles the same. Score Information: 2 The response to Part A includes no correct sequence of transformations (0), but a valid explanation of congruence (1). The response to Part B includes a valid explanation of similarity (1). 29
Anchor Paper #5 A. The trapezoids must be congruent because if trapezoid 1 is reflected across the y-axis and then reflected 8 units down, you see it is the same as trapezoid 2. B. Trapezoid 3 must be similar to trapezoid 1 even though the scale factor is unknown because the side lengths of the trapezoids are still proportional. Score Information: 1 The response to Part A includes an incorrect sequence of transformations (0) and no explanation of congruence (0). The response to Part B includes a valid explanation of similarity (1). 30
Anchor Paper #6 Part A: You move figure 2 up 8 units, then reflect it over the y-axis. They re congruent because the figure was not dialated in any way. Part B: It must be similar because it was dialated, the size was change, so the two figure are not congruent anymore. Score Information: 1 The response to Part A includes a correct sequence of transformations (mapping trapezoid 2 to trapezoid 1) (1), but an incomplete explanation of congruence (0). The response to Part B includes an incomplete explanation of similarity (0). 31
Anchor Paper #7 These two trapizoids must be congruent because if you take either one or the other, if you try to measure them up, they have to be the exact same. Trapizoid one and three must be similar because in order for any of them to be correct, they have to be similar in some way or form. Score Information: 0 The response to Part A includes no sequence of transformations (0) and an incorrect explanation of congruence (0). The response to Part B includes an incorrect explanation of similarity (0). 32
Anchor Paper #8 A.) The trapezoids must be congruent because they are the same size. They also have the same area. B.) Trapezoid 1 will be similar to trapezoid 3 because trapezoid 3 will be the same shape it will only get larger. Score Information: 0 The response to Part A includes no sequence of transformations (0) and an incomplete explanation of congruence (0). The response to Part B includes an incomplete explanation of similarity (0). 33
Mathematics Geometry, Item 55 Practice Scoring Exercise Five (5) sample responses have been selected and presented here to help scorers calibrate their expectations and judgments to ensure student responses are accurately and consistently scored. Scorers should: Review the rubric again. Read each bullet point and each score point descriptor carefully. Read each sample response. Give each sample response a score based on the rubric. Compare your scores with the key, noting any differences in how the responses were scored. Begin scoring student responses when confident that the rubric can be applied accurately and consistently. 34
Mathematics Geometry, Item 55, Practice Scoring Exercise Paper Score Justification for Score #1 #2 #3 #4 #5 35
Practice Paper #1 Part A: In order for trapezoid 1 to be congruent to trapezoid 2, the sequence must go from 90 to 180 to 270. The trapezoids must be congruent because of its similar slope. Part B: Trapezoid 3 must be similar to trapezoid 1 because it s a reflection of trapezoid 1. 36
Practice Paper #2 A) It flipped around B) There was no dialation 37
Practice Paper #3 Part A: Trapezoid 2 can be flipped across the y axis and moved up 8 units. That makes Trapezoid 2 on top of Trapezoid 1 and that shows they are the exact same size and shape so they are congruent. Part B: Trapezoid 3 must be similar to Trapezoid 1 because it s a dilation and dilations make the same shape. 38
Practice Paper #4 Part A Flipped over y-axis and then moved down 8 spaces. Part B It must be similar because they need similar coordinates in order to create a new trapezoid. 39
Practice Paper #5 Part A: 1) move down 8 2) flip across y-axis The sides and angles don t change, just the location. It s still the same size trapezoid but it just got moved a little. Part B: The sizes are still proportional after a dilation, it doesn t matter if the scale makes it bigger or smaller. 40
Mathematics Geometry, Item 55 Practice Scoring Exercise Key Paper Score Justification for Score #1 0 #2 0 #3 2 #4 1 #5 3 The response to Part A includes an incorrect sequence of transformations (0) and an incorrect explanation of congruence (0). The response to Part B includes an incorrect explanation of similarity (0). The response to Part A includes an incorrect sequence of transformations (0) and no explanation of congruence (0). The response to Part B includes an incorrect explanation of similarity (0). The response to Part A includes a correct sequence of transformations (1) and a valid explanation of congruence (1). The response to Part B includes an incorrect explanation of similarity (0). The response to Part A includes a correct sequence of transformations (1), but no explanation of congruence (0). The response to Part B includes an incorrect explanation of similarity (0). The response to Part A includes a correct sequence of transformations (1) and a valid explanation of congruence (1). The response to Part B includes a valid explanation of similarity (1). 41