Shape Hunting This problem gives you the chance to: identify and describe solid shapes

Transcription

1 Shape Hunting This problem gives you the chance to: identify and describe solid shapes Detective Sherlock Shapehunter tracks down solid shapes using clues provided by eyewitnesses. Here are some eyewitness reports. Which shapes do they describe? 1. This shape has 6 square faces. It does not roll. It is a regular prism. It has 8 vertices and 12 edges. This shape is a. 2. This shape has one curved surface that meets at a point. Its base is one flat circular surface. This shape is a. 3. This shape has one curved surface. It has 2 flat ends that are congruent circles. This shape is a. 4. An eyewitness saw this shape. Write three clues that describe it. The name of this shape is. Copyright 2007 by Mathematics Assessment Page 95 Shape Hunting Test 5

2 5. Write three clues that would help Detective Sherlock Shapehunter to track down a pyramid on a square base. 6. Detective Sherlock Shapehunter says that a sphere is different from all other solid shapes. Write two clues that describe how it is different from other shapes. 8 Copyright 2007 by Mathematics Assessment Page 96 Shape Hunting Test 5

3 Task 5: Shape Hunting Rubric The core elements of performance required by this task are: identify and describe solid shapes Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: cube 1 2. Gives correct answer: cone 1 3. Gives correct answer: cylinder Writes three correct clues such as: It has 6 vertices. It has 9 edges. It has 5 faces. 3 faces are rectangular. 2 faces are triangular. Gives a correct name: Triangular prism All four items correct. Partial credit 1 point for 2 or 3 correct items 2 (1) 2 5. Writes three correct clues such as: It has 5 faces. Four faces are congruent triangles. It has 5 vertices. It has 8 edges. It doesn t roll Partial credit 1 point for 2 correct clues. 2 (1) 2 6. Writes two correct clues such as: It has no flat surfaces. It has no edges. It has no vertices. It rolls in all directions. 1 1 Total Points 8 Copyright 2007 by Mathematics Assessment Page 97 Shape Hunting Test 5

4 Shape Hunting Work the task. Look at the rubric. What are the big mathematical ideas being tested in this task? Look at student work for parts 1-3. How many of your students: Could name all the shapes correctly? Used shape names of 2-dimensional shapes? Drew pictures to help them think about the shapes being described? Used non-academic descriptors, like box? Now look at the types of clues given for the triangular prism. How many of your students could: Could quantify the faces? edges? vertices? Used non-academic language: such as pointy or points, corners or sides Used base for the rectangular bottom of the shape instead of the parallel sides? Described the type of triangle, e.g. equilateral Were able to add adjectives, such as congruent, to describe additional information Discuss the parallel sides Show evidence of thinking at a low van Hiele level: e.g looks like a house, box, or tent; its long How many only counted 4 faces? 6 faces? 3 faces? Now look at clues for part 5. How many of your students said: Its pointy? Has 3 vertices? Has 3 sides? Gave story clues: bring a flashlight, used in games, hard to climb, holds mummies Separated out sides and bottom or 5 faces What opportunities have students had to work with 3-dimensional shapes this year? How do you promote discussions in the classroom that give students opportunities to use academic language for a purpose? Have students had opportunities to sort and classify 2-dimensional shapes? 3-dimensional shapes? Many students struggled with making representations of shapes being described. Do students in your classroom get frequent opportunities to draw figures for themselves? Do you provide activities that help them learn to make more accurate looking representations of 3-dimensional shapes? When describing shapes, do you talk about how rotating a shape does not change the base? Do you mention parallel in context of sides or faces? Do students in your class know that parallel can describe faces and planes as well as lines? Do you think your students understand what congruent means? What other ideas struck you as you looked at student work that you think students need to have clarified or that students need more experiences with? 5 th grade

5 Looking at Student Work on Shape Hunting Student A is an example of a student with full marks, whose clues show a high level of sophistication in geometric language and the attributes noticed. The student talks about parallel sides and uses words like congruent. Student A 5 th grade

6 Student A, part 2 Student B also has a well-developed geometric vocabulary and awareness of important geometric attributes to notice and quantify. Note the high-level explanation of the sphere in part 6. Student B 5 th grade

7 Student B, part 2 Many successful students made visual representations of the shapes to help them think about the properties being described. Student C also has a nice description for the sphere. Student D has full marks, but talks about the triangles as ends rather than parallel faces or bases. Student C 5 th grade

8 Student C, part 2 5 th grade

9 Student D Student E clearly thinks that the base is always the side on the bottom, rather than understanding that the shape can be rotated in space. What is a reasonable expectation for students at this grade level about this idea of base? How do you want to develop that idea over time to expand and deepen the understanding? Student E 5 th grade

10 Student F is able to name the shapes in on parts1-3. The students has 3 acceptable clues in part 4, but it would have been nice to describe what the congruent shapes looked like. How could you use this work to pose a question to the class that might make them see a need for given more details? In part 5 the student describes the pyramid as point on top? Is this a good geometric description? How is this vertex different? Why not 5 vertices? The student has a common misconception about the number of faces in the pyramid. Can you think of two or more reasons why students might give this answer? Student F As students start to get lower scores, they give 2-dimensional names to some of shapes. They also use more informal language to describe the attributes and don t give the level of detail. In part 5, Student G doesn t describe the shape of the sides or bottom, which are critical in order to identify the shape. Student G received 5 points on the task. 5 th grade

11 Student G Student H has a score of 2. The student is using some informal language, such as box and points. The student is unclear about critical attributes to describe, so instead talks about long without a reference. Notice that sides and bottom are different, as opposed to 5 faces. The student thinks the sphere has a base. What might the student be thinking? 5 th grade

12 Student H 5 th grade

13 At the lower end students have trouble drawing the geometric representations. Notice how Student I draws a curved surface with a point. 3 of the names used by Student I on page 1 of the task are of 2- dimensional shapes. Also the student incorrectly uses the word edge in part 5. What kinds of activities help students develop academic language? What opportunities have your students had that help them develop their ability to make better geometric representations? Student I 5 th grade

14 Student I, part 2 Student J also has difficulty with drawing the cube. Notice how the student describes the triangular prism. What van Hiele level is this student operating on? What is your evidence? Student J 5 th grade

15 Again, Student K has two points on this task. Look at how the representations. While the description for the sphere seems o.k. without the pictures, do you think the student really knows what a sphere is? Student K 5 th grade

16 The next set of work illustrates some of the difficulties students had with parts 5. 5 th grade

17 5 th grade

18 5 th Grade Task 5 Shape Hunting Student Task Core Idea 4 Geometry and Measurement Identify and describe solid shapes. Use geometrical attributes to describe a solid figure. Analyze characteristics and properties of two- and threedimensional geometric shapes, understand attributes, and apply appropriate techniques, tools, and formulas to determine measurements. Identify, compare, and analyze attributes of two- and threedimensional shapes, and develop vocabulary to describe the attributes. Based on teacher observation, this is what fifth graders knew and were able to do: Describe something about a sphere Identify a cube and a cylinder from clues Write two or three clues about a triangular prism Write two clues about a pyramid Areas of difficulty for fifth graders: Making representations of 3-dimensional shapes Using academic language to describe geometric attributes Recognizing a cone from clues Counting faces and vertices for 3-dimensional shapes Understanding the base of a prism as one of the parallel sides 5 th grade

19 The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, 88%, were able to either give two clues about a sphere, name the cube, or give 2 or 3 clues about the triangular prism. The clues may have used nonacademic language like sides and corners. More than half the students, 59%, can either name the first three shapes and give or two clues about the triangular prism or name the cylinder and give clues about the square pyramid and sphere. Almost 10% of the students could meet all the demands of the task including naming the triangular prism and giving clues about the triangular prism, the square pyramid, and the sphere. 12% of the students scored no points on the task. 70% of the students with this score attempted the task. Shape Hunting 5 th grade

20 Points Understandings Misunderstandings 0 70% of the students with this score attempted the task. Students used everyday language or incorrect language to describe the sphere: e.g. It looks like a ball. It s a circle. They also chose attributes that could be used for other shapes: 2 Students with this score could usually write two clues about a sphere and identify a cube from clues or write two or three clues about a triangular prism from looking at a diagram. 4 Students with this score could describe the sphere, recognize one or two shapes from descriptions, and write clues for either the triangular prism or the pyramid. Its round. 12% of the students thought the cube was a square. 10% thought the cube was a rectangle. 6% thought the cube was an octagon. 9% of the students described the overall shape for triangular prism, e.g. It looks like a roof or a box. 4% of the students said the triangular prism had a rectangular base. 3% said that it had 6 faces and another 3% thought it had 3 faces. 6% of the students only gave two clues for the triangular prism. 5% of the students did not attempt to write clues for the triangular prism. 7.6% of the students thought the cone was a sphere. 8% thought the cone was a circle. 6% did not attempt to identify the shape described in part 2. 8% thought the cylinder was a sphere. 6% thought the cylinder was a circle. 7% did not attempt to name the cylinder from the clues. Students couldn t name the triangular prism. 12% left the name blank. 5% called it a triangle. 6 Students with this score generally could not write clues for the pyramid. 10% said it looks like a triangle. 12% used nonmathematical clues: Its found on money. Bring a flashlight. 12% said it had 4 sides or faces. 4% thought it had 3 faces. 8% said there were 3 triangles. 10% said its pointy. 8 Students could name 3-dimensional shapes from clues. They could write clues for a triangular prism given a diagram. They could write clues for a sphere and square pyramid without drawings. They new important attributes for describing 3-dimensional shapes. 5 th grade

21 Implications for Instruction Students need to be able to describe attributes of 2- and 3- dimensional shapes. They should be able to describe the shape by describing the number of faces, number of vertices, and number of edges. They should begin to think about how shapes move in space, so that they can think about the base of a prism as not the bottom of a shape, but one of the parallel faces. They should be familiar with a variety of 3-dimensional shapes including cones, cylinders, spheres, prisms, and pyramids. Students develop academic language by having opportunities to use language with a purpose. So activities where students write and exchange clues and then have to guess the shape being described would help students focus on language. Putting a variety of shapes in a bag and having students reach in and try to describe the attributes of the shape by feel, helps students to think less about the overall shape (i.e. it looks like a house) to attributes of the shape. Students should also have opportunities to practice drawing or sketching 3-dimensional shapes. Being able to record ideas helps students to notice more of the details or attributes of the shapes. This ability to visualize and record the shape also aid in counting edges, faces and vertices. Students find it very empowering to learn these skills. Ideas for Action Research Incorporating Geometry Warm Ups Consider ways to plan geometry warm ups or fillers into your plans. Think about how to incorporate drawing or sketching on a routine basis instead of waiting for the end of the year when you teach geometry. Plan some attribute games that can be used when you have a few extra minutes, like feeling shapes in the bag and giving attributes. Use geometric shapes for developing writing skills or higher order thinking skills. Make worksheets with a couple of shapes for students to describe similarities and differences or just hold up shapes that you want students to write about. Building shapes with linker cubes and then trying to sketch on triangular dot paper is a nice way to settle students down after that have been over-stimulated by lunch, assemblies, etc. Show them a shape, like an ell made of 4 blocks. Ask them to build it and then draw it. For a longer or more extended activity, give students some 3- dimensional shapes to sort. Allow students to develop their categories for sorting the objects. Students will sort by the attributes they are most comfortable with. See if groups can guess the rules they used for sorting. Reflecting on the Results for Fifth Grade as a Whole Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? 5 th grade

22 If you were to describe one or two big ideas to take away and use for planning for next year, what would they be? What are some of the qualities that you saw in good work or strategies used by good students that you would like to help other students develop? Four areas that stand out for the Collaborative as a whole are: 1. Multiplicative Thinking Students need to be able to think about a group of various sizes and how to reiterate or grow that group. In Candies, students are asked to think about ratios and how the quantity of individual items would change if the total number of items were increased. They needed to think about how many of units of 5 would fit into thirty. Understanding the language of groups and numbers of groups will help develop their understanding. In Joyce s Rugs students needed to think about circles growing in groups of 2 and hopefully express that relationship with multiplication. In Granny s Balloon, students needed to think about making equal-size groups or intervals when designing the scale for the vertical axis. Students also needed to reason proportionally about marks falling between the grid lines or units to estimate time and distance. 2. Choosing Operations Students had difficulty choosing the appropriate operations for Cindy s Cats. They need models to help them make sense of the action of the problem. Students also had difficulty tracking the meaning of their answers after they had made calculations. Labeling helps students to make sense of operations or understand why an operation doesn t make sense. More attention to labels might have helped students in both Candies and Cindy s Cats. 3. Cognitive Demands of Tasks There is a significant difference in the cognitive demands of making a graph than reading information off a graph. The number of details that students must consider and incorporate and very different. The same is true for using a table. When filling in a table that is given to them, students often lose track of context and relationships. They fill in comfortable or familiar patterns, often not picking up on all the given clues. When students make their own tables as a strategy to help them think, they are more likely to do it correctly and use it for sense-making. 4. Spatial Visualization and Representation Students need experience with making their own representations of objects they see. In recording 2 and 3 dimensional shapes for themselves they are more likely to focus in on key mathematical attributes of the shape. Research shows that maturation does not increase ability to discern attributes. Experience is the only way to move through the stages of spatial thinking. 5 th grade