Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 inches. 24 inches

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1 Winter Hat This problem gives you the chance to: calculate the dimensions of material needed for a hat use circle, circumference and area, trapezoid and rectangle Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches 3 inches 3.5 inches x inches Circumference of circle =πd Area of circle = πr 2 24 inches 2.5 inches 1. The rectangular strip is 24 inches long. Eight trapezoids fit together around the rectangular strip. Find the width (x) of the base of each trapezoid inches 2. The circle at the top of the hat has a diameter of 3 inches. a. Find the circumference of the circle. Show your calculation. inches b. Eight trapezoids fit around the circle. Find the width (y) of the top of each trapezoid? inches 3. Find the surface area of the outside of the hat. Show all your calculations. square inches 9 Copyright 2008 by Mathematics Assessment Resource Service 60

2 Winter Hat The core elements of performance required by this task are: calculate the dimensions of material needed for a hat use circle, circumference and area, trapezoid and rectangle Based on these, credit for specific aspects of performance should be assigned as follows Rubric points section points 1. Gives correct answer: 3 inches 1 2.a. b. Gives correct answer: 9.4 or 3π inches Shows correct work such as: π x 3 Gives correct answer: 1.2 or 3 / 8 π inches 3. Gives correct answer: 126 square inches Allow 125 to 129 Shows correct work such as: 24 x 2.5 = 60 (rectangle) π x = 2.25 π = 7.1 (circle) ( ) / 2 x 3.5 = 7.35 (trapezoid) 7.35 x 8 = 58.8 (8 trapezoids) 1 1 1ft ft 1ft 5 Total Points Copyright 2008 by Mathematics Assessment Resource Service 61

3 Winter Hat Work the task. Look at the rubric. What are the mathematical concepts being assessed in this task? Look at student work for part 2b, finding the width of the top of each trapezoid using the circumference of the small circle. How many of your students put: or # 15 Other What is some of the thinking behind these misconceptions? What might the students with answers of 1.1 or 1 been thinking? How is this misconception different from that of students with answers of 3 or 3.5? Now look at work for part 3. How many of your students: Labeled calculations so they knew which was the area of the rectangle, the area of the trapezoid, etc.? Correctly found the area of the rectangle? Correctly found the area of the circle? Correctly found the area of a trapezoid? Tried to find the area of a trapezoid but used an incorrect formula? Tried to find the area of 8 trapezoids? Multiplied areas of different figures together? Used dimensions from different figures in attempting to find area? Found perimeter of shapes? Struggled to interpret the diagram of the hat? How often are students in your class asked to do a task with a long reasoning chain? How often do students solve problems where they need to compute something to use as dimension for something else? Look in your textbooks. What opportunities do students have to interpret complex diagrams? How much more practice is devoted to computation devoid of diagrams, where the measurements are just given? How is the thinking and understanding significantly different in these two situations? 62

4 What opportunities have students had making nets and unfolding them to look at the individual pieces? What do you think students understand about the process of finding surface area? Looking at Student Work on Winter Hat Student A uses labels and units to organize work. Notice how the student makes new diagrams for the shapes and labels the dimensions in order to think through the calculations in part 3. How do we help students develop this habit of mind? Student A 63

5 Student B is also able to organize the work in part 3, using diagrams to label the calculations. Notice the student does not round off numbers. How do we help students to make sense of numbers from calculators? In making a pattern would it make sense to try for this level of accuracy? Student B 64

6 Student C does not label or organize the work. The student understands that surface area is the total area of all the pattern parts. In part 3 the student can calculate the area of the rectangle and the small circle. The student does not know the formula for trapezoid (4 th and 5 th grade standard) and finds half of one base rather than half the total of the 2 bases. The student forgets that there are 8 trapezoids. Student C 65

7 Student D has trouble interpreting the diagram. The student is able to find the circumference in the small circle in 2a. However when thinking about fitting the 8 trapezoids around the circle, the student divides the diameter by 8 instead of using the circumference. The student is able to find the area of the rectangle. The student uses the formula for area of rectangle instead of area of a trapezoid, but does know that there are 8 trapezoids. The student doesn t square the radius when finding the area of the circle. Again the student does not think about significant digits in the final answer to 3. Student D 66

8 Student E is able to calculate the width of the trapezoid and the diameter of the small circle. She divides the circumference by 8 to find the width of the top of the trapezoid (1.17) but rounds incorrectly. In part 3 the student only calculates the area of the circle. The student does not think about surface area as the sum of all the sides. Student E 67

9 Student F is good at calculations, when told explicitly what to find. Notice that in part three the student breaks the trapezoid into a rectangle (moving one triangle to the other side) to calculate the area for the trapezoid. It is unclear how the student decided on the size of the base (4) or if that is a rounded number (3.5 4). The student adds in the circumference to the area of the other shapes. What types of experiences would help this student? What questions might you ask? Student F 68

10 Student G calculates the circumference of the small circle and uses that as one of the dimensions of the large rectangle, replacing the 2.5. Do students get enough opportunities to think about and deconstruct diagrams as part of their regular class program? How do we help students develop their visual thinking? Again, the student struggles with interpreting the diagram when thinking about 2b. The student thinks now about using the circumference of the circle as the top dimension of the trapezoid. Would a habit of mind, like labeling diagrams with dimensions, have helped this student? Why or why not? Finally in part 3 the student multiplies the width of the trapezoid by 8 instead of the area of the trapezoid. The student adds this calculation to the other top side of the trapezoid. There is no use of area in any of the calculations in part 3. Student G 69

11 Student H tries to find the area of the rectangle in part one, but struggles with multiplying decimals. How has the student dealt with multiplying by 0.5? So the student then divides the area by 8 instead of the circumference by 8. In part 2a the student finds the radius instead of the circumference and uses that as a dimension of the trapezoid. Is this student struggling with understanding the diagram? What other issues are at play? In part 3 the student uses the derived width of the trapezoid times 8 rather than multiplying an area times 8. Would labels help this student? What experiences might help the student make sense of the context and what is being asked? Student H 70

12 Student I appears to multiply pi by the length of the rectangle rather than the diameter of the small circle in part 2a. The student has a correct answer for 2b, but there is no supporting work. In part 3 the student calculates the area of the rectangle in the work above the question. The student then also calculates the area of the circle (again in the work above the prompt for 3.) The student also appears to have an area for the trapezoid, but doesn t use it below. In the final work the student seems to multiply the area of the circle times the eight trapezoids and the circle), but then doesn t use that calculation. The final total could be either the area of the rectangle and the area of the circle or the area of the rectangle and the area of the trapezoid. Students need to have practice organizing large tasks for themselves to develop the logic of tracking calculations. Students also need to see and compare examples of how to organize work in order to improve their skills. Student I 71

13 Student J does not understand the demands of the task. Most answers have no attached calculations or the student uses the dimensions from part of the diagram. In part 3 the student appears to have measured the picture of the hat and used those dimensions to find the perimeter instead of thinking about surface area. What resources are currently available at your school site to help students who are missing this much background knowledge? What are reasonable steps you can take within the classroom? How can you help the student get other services? Student J 72

14 7 th Grade Task 4 Winter Hat Student Task Core Idea 4 Geometry and Measurement Calculate the dimensions of material needed for a hat. Use circle, measures of circumference and area. Calculate area for rectangles and trapezoids. Analyze characteristics and properties of two-dimensional geometric shapes. Apply appropriate techniques, tools, and formulas to determine measurements. Develop, understand, and use formulas to determine area of quadrilaterals and the circumference and area of circles. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes. Select and apply techniques and tools to accurately find length, area, and angle measures to appropriate levels of precision. The mathematics of this task: Calculating geometric units, such as area, circumference, radius, surface area Composing and decomposing 3-dimensional shapes Diagram literacy, being able to read and interpret a diagram and match parts of the diagram to given dimensions, such as, seeing how circumference relates to the rectangular shape, understanding what parts of the trapezoids connect to other parts of the figure Based on teacher observations, this is what seventh graders know and are able to do: Calculate the circumference of a circle Divide 24 by 8 to get the width of the trapezoid Give a value for pi Areas of difficulty for seventh graders: Knowing the formula for the area of a trapezoid Understanding what dimensions or measurements are needed to find the area of a trapezoid Visualizing how the sides of the trapezoid connect to the rest of the diagram Confusing area and circumference of a circle Understanding a diagram and breaking it down into separate parts Understanding of how to find surface area Organizing work to keep track of what is known, what is being calculated, what else needs to be calculated Strategies used by successful students: Labeling answers and defining what is being calculated each time Writing dimensions on the diagram as they are calculated for quick reference for future parts of the task 73

15 The maximum score available for this task is 9 points. The minimum score needed for a level 3 response, meeting standard, is 4 points. Many students, about 77%, could divide the width of the rectangle by 8 to find the bottom dimension for the trapezoids. More than half the students, 50%, could also calculate the circumference of the small circle and show their calculations. Some students, 35%, could also divide the circumference of the small circle by 8 to find the top dimension for the trapezoid. Less than 3% of the students could meet all the demands of the task including finding the surface area of a 3-dimensional shape composed of a rectangle, a circle, and 8 trapezoids. 37% of the students scored no points on this task. 60% of the students with this score attempted the task. 74

16 Winter Hat Points Understandings Misunderstandings 0 Only 60% of the students with this score attempted the task. Students could not reason about how the trapezoids attached to the rectangle. Answers for part 1 ranged from 1.5 to 1 Students could divide the width of the long rectangle by 8 to find the width of the base of the trapezoid. 2 Students with this score could usually find the circumference of the small circle and show their work. (These students could not find the answer in part 1.) 4 Students could solve parts 1 and 2a, showing their work. 7 Students could use the given measurements to find the area of the rectangle, the dimensions of the trapezoid, and find the circumference of the small circle. They understood that surface area meant adding together the parts. 9 Students could reason about a complex 3-dimensional shape, using a series of calculations to derive needed dimensions, and using the dimensions to calculate surface area The most common error was 3.5 Students had difficulty finding the circumference of the small circle. Common errors were 6,9, 1.5, and 5.5. Students struggled with decomposing shape to understand the relationship between the circumference and the top of the trapezoid. Some students did not round properly (3%). 9% found radius instead of circumference. 3% gave the height of the trapezoid instead of the length of the top. Students had difficulty organizing their thinking to find surface area of a complex shape. 15% did not attempt to find the area of a rectangle. Many students did not think about the fact that 8 trapezoids were needed to make the complete hat. (74%) Students struggled with the longer reasoning chain and organizing their thinking. Students with this score could not find the area of the circle. 54% did not attempt to find the area of a circle. 10% used the circumference instead of the area to add with the other shapes. 3% just squared the radius and forgot to multiply by pi. Students did not remember or could not use the formula for area of a trapezoid. 43% did not attempt to make this calculation. 11% multiplied the lower base by the height. 6% just used the height for the area. 75

17 Implications for Instruction Students at this grade level should be challenged frequently to work on larger problems involving longer chains of reasoning. Students have been working with geometric concepts, such as area and perimeter of rectangles since 3 rd grade, area of trapezoids in 5 th grade, and area of circles in 5 th and 6 th grade. The challenge for this grade level is to think about complex shapes and geometric relationships. Students need to learn to organize their thinking and do their own scaffolding. Students should develop tools and habits of mind for making sense of what they know and what they need to find out. Using labels for their calculations can help them think through the reasoning process. Students had trouble making sense of diagrams. Students need more concrete experiences with building and decomposing 3-dimensional shapes to help them think about moving between a figure drawing and a net. Students at all grade levels seem to fear writing on diagrams. This can be a powerful tool to aid in thinking. Writing dimensions directly on the diagrams helps students track their thinking and plan what needs to come next. To prepare students for algebraic thinking, students at this grade level should start to make generalizations about geometric formulas and understand how they are derived. Instead of memorizing lots of different formulas, students should look at the trapezoid and think about averaging the two bases to make a rectangle. Ideas for Action Research Problems of the Month One interesting task to help students stretch their thinking about 3-dimensional shapes is the problem of the month: Piece it Together, from the Noyce Website: Ask students to work individually or in teams to solve the problem. Have them make posters of calculations they made, their conclusions, and graphics or visuals to support their thinking. The poster might also include other ideas they want to explore or conjectures they haven t had time to test. The purpose is to give them some complex mathematical thinking, that requires persistence, willingness to make mistakes, edit and revise, and is worth understanding the thinking of others. 76

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