(6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.
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1 (6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. (6.6.a) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to use angle measurements to classify angles as acute, obtuse, or right. Clarifying Activity with Students measure angles found in objects around the room and label the angles with the appropriate classification (acute, obtuse, or right). Open with... What determined the label you chose for each angle? How many categories did you use? What makes an angle acute? Right? Obtuse? How did you use visual clues to identify each angle? Did you find examples for all angle classifications? Can an angle be in more than one category? Why? What helped you categorize the angles accurately? Is one type of angle represented more often than another type of angle? Or less? Why? Is there a connection between an object's angles and its shape? Can you justify your answer? Do the object's angles relate to the design of the structure? If so, how? What do the angles tell us about the object's function? How can categorizing angles help you estimate angle measurement in the real world? Does the student use definitions for acute, obtuse, and right angles correctly? Can the student justify his or her choice of labels? Is the student using visual clues to categorize the angles before measuring? Can the student find angles in objects? Can the student use tools appropriately to measure angles? Can the student use organizational models to record and sort angles?
2 Can the student identify the rays of the angles? Spring 2003, grade 6, item 4 Spring 2006, grade 6, items 41 and 44 (6.6.b) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to identify relationships involving angles in triangles and quadrilaterals. Clarifying Activity with Given a set of different triangles or quadrilaterals, students measure the angles (using a protractor or geometry exploration software) and classify polygons according to their properties. Students organize their data in a table and analyze patterns to match triangles or quadrilaterals with the most appropriate polygon name. Open with... Describe the relationship between the angle measurements and the specific name of the triangle or quadrilateral. Do you see patterns in the angle measurements of a triangle? Quadrilateral? Do you see similarities and differences between the polygons and their angles? Can the definitions of acute, obtuse, or right angles help you sort the polygons by name? Do any triangles or quadrilaterals have two congruent angles? Three congruent angles? All congruent angles? Are there ways to organize your data to help you determine patterns? Is the student using geometric terms to analyze the properties of each polygon? Does the student use the definitions for acute, obtuse, and right angles correctly? Is the student making conjectures about the relationships between angles and polygons? Is the student comparing the angle measurements?
3 Is the student generalizing accurately concerning the attributes of the different types of triangles? Quadrilaterals? Is the student extending the angle-to-polygon analysis to specific triangle names? Specific quadrilateral names? Is the student measuring the angles accurately? Is the student looking for patterns? Is the student representing the sorted polygons with the best geometric labels? Can the student categorize the polygons according to the attributes seen? Can the student justify the categories using specific properties of angles? Is the student checking polygon arrangements for errors or duplications? Can the student extend angle-to-polygon relationships from one category to another? Can the student organize his or her information in a table? Spring 2003, grade 6, item 30 Spring 2006, grade 6, items 9 and 38 (6.6.c) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to describe the relationship between radius, diameter, and circumference of a circle. Clarifying Activity with Students use tape measures or string to measure the radii, diameters, and circumferences of several circular objects, organize their data in a table, and look for patterns and relationships using a calculator. Students answer questions such as "What is the relationship between the length of the radius and the length of the diameter? The length of the diameter and the circumference?" (For example, "The length of the diameter is twice the length of the radius.") Students then use other circular objects to test the patterns and relationships found. For example, students could measure circumference of a circular object to be, for instance, 12.5 inches, and then use the relationships they found to calculate the diameter. Then students measure the diameter and compare the measurement with their calculation.
4 Open with... What are the relationships between radius, diameter, and circumference? How can you find the circumference of a circle? How can you find the diameter of a circle if you know the radius? How can you find the radius of a circle if you know the diameter? Do you see any patterns in the measurements that might help you predict radius, diameter, and circumference? If you know the diameter, how can you find the circumference? If you know the circumference, how can you find the diameter? How did your calculation of the diameter for the circle with the circumference that you found compare with the measurement you made of the diameter? Can the student verbalize patterns? Is the student making reasonable connections between the radius, diameter, and circumference of a circle? Does the student use appropriate vocabulary? Can the student hypothesize relationships? Does the student use the calculator correctly? Is the student using ratios to represent relationships? Is the student measuring radius, diameter, and circumference with matching units of measure? Does the student use a unit of measure appropriate for the size of the object (e.g., feet for large objects and inches for small)? Is the student looking for multiplicative patterns? Is the student organizing his or her data and looking for patterns? Is the student recording his or her findings? Is the student using different strategies to measure? Is the student checking for reasonableness of calculations? Spring 2003, grade 6, item 27 Spring 2004, grade 6, item 23 Spring 2006, grade 6, item 13
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