Marie has a winter hat made from a circle, a rectangular strip and eight trapezoid shaped pieces. y inches. 3 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 Grade 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 Grade 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? Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 Grade
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 twodimensional 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 3dimensional 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 Grade
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 3dimensional 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. Grade
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 3dimensional 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. Grade
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 3dimensional 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 3dimensional 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. Grade
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22 Sale! This problem gives you the chance to: work with sales discount offers and percents The following price reductions are available. Two for the price of one Buy one and get 25% off the second Buy two and get 50% off the second one Three for the price of two 1. Which of these four different offers gives the biggest price reduction? Explain your reasoning clearly. 2. Which of these four different offers gives the smallest price reduction? Explain your reasoning clearly. 9 Grade Copyright 2008 by Mathematics Assessment Resource Service 81
23 Sale! Rubric The core elements of performance required by this task are: work with sales discount offers and percents Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer Two for the price of one. Gives an explanation distinguishing which is the best buy. Ranks all items by sample cost per item, % reduction per item, or fractional cost per item, such as: If the original price of one item is $100, then Two for the price of one means that each item costs $50 or 50% of the original price. Buy one and get 25% off the second means that each item costs $87.50 or 87.5% of the original price Buy two and get 50% off the second means that each item costs $75 or 75% of the original price Three for the price of two means that each item costs $66.67 or 66.7% of the original price 2. Gives correct answer: Buy one and get 25% off the second Gives an explanation distinguishing between the two lowest reductions or explains why this is the worst choice. 1 3 Total Points Grade Copyright 2008 by Mathematics Assessment Resource Service 82
24 Sale! Work the task and look at the rubric. How did you rank the 4 options before making the comparison? What unit did you choose to make the comparison? What are some other units that could be used? Look at student work for part 1. How many of your students chose : 2 for the price of 1 Buy 1 get 25% off the second Buy 1 get 50% off the second 3 for the price of 2 Thought that 2 for 1 and 3 for 2 were equally good. How many students were able to: Rank all the choices using some common unit to make the comparison Ranked at least 3 of the 4 choices with a common unit Gave a fair justification distinguishing between the top two choices Gave an explanation only involving their choice Gave an incorrect comparison because of lack of understanding of the situation or incorrect conclusion Make a list of the types of reasons or justifications students used for their choice: Did students think about changing the options to a similar unit before comparing? Did students understand the difference between getting one free in the first answer and one free in the second? How did they justify or quantify the advantage? Did any of your students think about a unit price? What were some of the options that didn t make sense? What are some of the errors students made in working with percents? What evidence did you find that students couldn t make sense of the different options? How often do students get an opportunity to work with problems like this, or rate problems, where it is important to make units the same to make a comparison? Why is this important? How has your class worked with the idea of making mathematical comparisons? How often do students work with problems where they need to make derived calculations before making a reasoned choice? What are the mathematical norms about using mathematics to quantify or justify a decision? Grade
25 When thinking about providing a balanced curriculum, students need to be exposed to a variety of task types; including: open investigation, nonroutine problems, design, plan, evaluation and recommendation, review and critique, representation of information, technical exercise, and definition of concepts. This task focuses on the evaluation and recommendation. Now look at student work on part 2. How many of your students chose: Buy one and get 25% off the second Two for the price of one Buy two and get 50% off the second Three for the price of two What were some of the reasons for their errors? What misconceptions did you see? Now look at student work for part 2 of the task. How many of your students chose: Buy 1 get 25% off the second 3 for the price of 2 2 for the price of 1 Buy 1 get 50% off the second Make a list of some of the units used by successful students to rank the choices: What were some of the most indefensible explanations: Grade
26 Looking at Student Work on Sale! Student A is able to think about the comparison by using a powerful mathematical strategy of pretending, what if the original price was $10. The student is then able to find the unit price for each shirt for all of the four options. The relationship between options remains the same no matter what the pretend price is. Student A Grade
27 Student B is able to compare three of the choices by finding the cost for 2 items. The student was able to eliminate the final option by comparing the price for 6 items between the best buy from the first three and comparing it to the remaining option. In both sets of comparisons the student fixes the number of items to compare costs. Student B Grade
28 Student C also fixes the number of items to compare the costs of the options. The student chooses the best buy from the first three options and then compares it to the final option. Notice that even though the student chooses a different pretend price and buys a different amount of shirts, the relationship between options stays the same. Student C Grade
29 Instead of comparing money spent, Student D compares money saved. The student runs into trouble on making the final comparison because he doesn t compare the savings on buying the same number of shirts. Student D Grade
30 The student correctly orders the options. The student shows that the first choice is 2 items for $10 and the 4th choice is 3 for $20, so neither the price nor the quantity is fixed. The student then makes a comparison between option 2 and 3, but does not quantify why they are a worse buy than option 1 or 4. Student E Grade
31 Student F is able to show a comparison between option 1,2, and 3 by finding the total cost of 2 items. The student chooses the wrong option because he does not convert the final option into the cost of two items. The student forgets that $3 is less than $6. If the student had fixed the cost: For $6 option 1 will get 4 items and option 4 will only get 3 items. If the student fixes the number of items, option 1 will get 2 items for $3 and option for will get 2 items for $4. Student F Grade
32 In part one Student G only talks about the good features of the best buy. The student makes no comparison to the other options. Students need to learn the structure of comparison and making a convincing argument. In part 2 the student is able to think about cost between the choice in one and the choice in two. The options are still not eliminated. How do we provide students opportunities to listen to convincing arguments or develop their own convincing arguments? How do they learn the logic of making a sound argument? Student G Grade
33 Student H has pretended the cost is $2. The student then finds the cost of buying items with each option. The problem is that the student is comparing buying 2 items in the first three amounts and the cost of 3 items in the final option. When making a comparison either the amount of items or the money spent needs to be equivalent before making the comparison. Student H Grade
34 Student I might be thinking that the cost for each item is $1, although that is not clearly stated. While the logic seems reasonable on first read for part, if the student quantified the options by comparing the same number of items she would see the error in thinking. Student I Grade
35 Student K has a strategy that could have led to the correct answer. Because the student is trying to do the calculations in his head, he may be losing track of the amounts. For example if the item is $40, then for option 2 he should have subtracted $10 from the cost rather than adding $10 to the cost. It doesn t get more expensive. What should the costs be for the other options? Student K Grade
36 In part 1 and part 2, Student L only justifies the qualities of the option picked. The student does not understand the language and logic of a comparison. What opportunities do you provide for students in your classroom to help develop these thinking skills? Student L Grade
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39 7 th Grade Task 5 Sale! Student Task Core Idea 1 Number and Operation Work with sales discount offers and percents. Make mathematical comparisons of different options by fixing one quantity. Develop, analyze and explain methods for solving problems involving proportional reasoning, such as scaling and finding equivalent ratios. Work flexibly with fractions, decimals, and percents to solve problems. Understand the meaning and effects of operations with rational numbers. Mathematics of the task: Making equivalent units for comparison Understanding the logic of comparison Using percents, fractions, etc. to find cost Based on teacher observation, this is what seventh graders knew and were able to do: Getting something for free 25% is less than 50%, and could find 50% of a number Almost everyone could pick the best buy Understood that two for the price of one means buy one and get one free Areas of difficulty for seventh graders: Making all four options in comparable units, many could change the first 3 into equivalent units, but didn t understand how to deal with the extra shirt in the final option Did not do calculations or confused percent off with percent paid Thought buy two and get 50% off second one means you re buying three items Understanding the logic of a comparison: one quantity needs to be fixed to make the comparison. In this case the students could either fix the number of items or fix the amount paid Strategies used by successful students: Using numerical values to determine the discounts or price paid Make up a starting cost for the items to quantify sales price or discount Using common multiples to set the number of items the same Grade
40 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, 81%, could identify either the best or the worst option for buying the items. Less than half the students, 45%, could identify both the best and the worst option. A few students, 19%, could also give a reason for their choices. 7% of the students could meet all the demands of the task including ranking all the options using equivalent units for comparison. Almost 19% scored no points on this task. 73% of those students attempted the task. Grade
41 Sale! Points Understandings Misunderstandings 0 73% of the students with this score attempted the task. Students were confused by the number of shirts. 19% thought getting 3 shirts was a better deal than buy one get one free. 13% 2 Students could identify the best and worst buy. 3 Students with this score could identify the worst buy and give a reason why it was worse than the next lowest option. 4 Students could identify both the best and the worst option. 6 Students could identify best and worst options and give some supporting reasons for their choices. 9 Students could find a way to rank the options using equivalent units and use this ranking to chose the best and worst buys. Students understood the idea of fixing either the number of items or the price to make the comparison. Most commonly students compared the first three options, then made a different unit to compare the best of those choices to the final choice. Many students understood that they could pretend the original price. They then could compare total cost, savings, fractional or percent amount paid. thought 50% off was the best deal. Students could not make convincing arguments to support their ideas. Many students only made comments about the option they chose, not even attempting a comparison with the other options. 15% of the students though 3 for the price of 2 was the worst option. 10% thought 50% off was the worst option. Almost 6% thought 2 for the price of one was the worst option. Students could not give convincing reasons for their choices. Students did not rank all the options by making some kind of equivalent unit. 69% of the students did not attempt to rank the solutions. 10% attempted to rank the items but didn t know how to deal with the three for the price of 2. Grade
42 Implications for Instruction Students need more practice reasoning about percents and quantifying relationships to make comparisons between different options. Many students could pick out the best or worst option, but gave little or no justification for their choice. They don t understand the logic or language of making a comparison. Students need to be able to find a common unit for comparison, such as a fraction, the money saved, or the total cost for buying the same quantity to help verify their choice. Students need to be able to fix one quantity in order to compare it to the other options. Ideas for Action Research Writing a Reengagement Lesson One of the effective tools developed by MAC through Action Research is the idea of reengagement, or using student work to plan further lessons. After the initial task the teacher or a group of teachers sits down to examine student work. As teachers read several papers, a story emerges. Sometimes the story is about strategies used by successful students. Sometimes the story is about the different demands in working with a concept in context versus being given the information or data. Other times the story is about a common recurring misconception. Sometimes the story is about the difference in cognitive demand in the task, such as finding derived measurements in Winter Hat or finding height in last year s Parallelogram task, and how students are presented with similar tasks in their textbooks. Sometimes the story is about recognizing significant digits, understanding remainders, understanding a big mathematical idea, such as what an average does or how to make a comparison. In this part of the tool kit, a larger sample of student work has been provided to give you an opportunity to develop your own reengagement lesson. Sit down with a group of colleagues to examine the work on Sale! or look at your own student work. What are the important mathematical ideas that students are struggling with in this task? What do you think is the story of this task? Now think, what is the minimum that most students in the class can understand and think about a series of 3 to 5 questions to help move student thinking to the more desired goal. What pieces of student work could you use to pose the questions? It helps to personalize the questions by attaching student names (not names from your class, because you don t want to put someone on the spot) or by posing a dilemma to get students intrigued about finding the solution. For example: Julie says that two for the price of three is a better deal than two for the price of one. Because in both cases you get one free item, but in her choice you get more items. Do you think Julie is correct? Why or Why not? Or Ken says suppose the item was a video game, which cost $40. Then buy 1 get 50% off would be $60. Can Ken pick a price? What would be the price for the other options? Would the best buy be the same if Ken had picked a different price? Grade
43 Or If the items cost $9 each, Martha says that 3 for the price of 2 would cost $18. Juanita thinks it would cost $27. Diedra says the price per item would be $6. What are they thinking? Who do you think is correct? When using student work, it is important not to show too much or use labels. The purpose is to make all students in the class have a reason or rethink the mathematics from a slightly different perspective. For example, a teacher might make a poster (without the correction mark) of Tabitha s work. What do you think this student was thinking? Where do the numbers come from? What are the figures in the drawing? Hopefully, class discussion will get students to think about each item originally costing $1. The common misconception that the option buy 2 and get 50% off the second is about 3 items will be surfaced. Also it can bring into play how to compare buying 2 items versus 3 items. Students or the teacher may then pose a further question about what would happen to the price if we bought the same number of items for each option. This type of minimal student work, or naked question, provokes thinking by all students. The cognitive demand of trying to get into someone else s head is engaging and much deeper than just solving the problem originally. The purpose of the reengagement lesson is not to give students another worksheet, but provide a forum for the exchange of ideas so that students develop the logic of making convincing arguments, confront their misconceptions explicitly, learn strategies used by others through the feedback of listening to their peers. Now try writing or designing a series of questions that you might pose for your class to develop the important mathematics of this task. Grade
44 Reflecting on the Results for Seventh 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? 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. Increasing the Cognitive Demands and Rigor  As students move through the grades the expectations for thinking about a topic and the mathematics they use should deepen and become more complex. In early grades, usually fourth, students learn to use words like likely, unlikely, certain to describe probabilities. They also learn to write numerical probabilities for simple events like drawing a marble out of a bag or rolling a die. What is new and different for middle grades is to be able to think about the sample space for compound events and be able to quantify the numerical probability. Students had difficulty with this in Will it Happen? At third and fourth grade students are learning to draw and extend patterns and work backwards from a total to a stage number. What is new and different for middle grades is the idea of searching for generalizations. Students should not need to draw every shape or add on in long strings to find an answer. They should be letting go of this comfortable habit and learning some new strategies.(odd Numbers) In geometry students have been working with area and perimeter since third grade. At this level students need to work with longer chains of reasoning, where it is necessary to look at geometric relationships to derive some of the measurements or dimensions needed for achieving the final solution. They need to be able to relate their calculations to specific parts of the diagram to reason about the problem solution. Students should also be able to think about the relationships involved in how formulas are put together (generalization), so they should be able to understand why the 2 bases of the trapezoid need to be added together and averaged. (Winter Hat) 2. Using Academic Language and Number Theory Students struggled with academic language and number theory in both Odd Numbers and Pedro s Tables. Students did not recognize categories of numbers, such as multiples of 6, consecutive odd numbers, Grade
45 square numbers, and prime numbers. Students confused factors and multiples. Students need opportunities to use academic language in conversation for a purpose, like making a convincing argument or explaining their thinking. Language acquisition is developed with practice. In probability students confused the mathematical idea of likely (having a more than 50% chance of happening) with the common language of likely(it is possible that it could happen). 3. Logic of Making a Justification In Pedro s Table students had difficulties justifying why multiples of 5 were not factors of 12 or why there was only one factor of 3 that was a prime number. Students didn t know how to tie facts or information back to the original assertion or how to connect different parts of the argument into a cohesive whole. In Sale! Students did not understand the format or language of making a comparison. Students might choose an option and explain why it is great without any references to the other choices. Students didn t understand that to make a comparison, there needs to be one quantity that is fixed so the other quantities can be compared. In this case either the number of items purchased needed to be the same so that the cost, the savings, or unit price could be compared or the cost needed to be fixed so that the number of items could be compared. 4. Diagram Literacy Students did not know how to make models for sample space to help them reason about compound probability in Will It Happen?. Students did not understand how the labels for rows and columns worked together in the table for Pedro s Table. Students could not think about how to decompose the diagram of the hat in Winter Hat, to think about the individual pieces. The students could not visualize how the trapezoid connected to the large rectangle or the circumference of the small circle. Students did not feel comfortable writing dimensions on the diagram as they were derived to help them think about the relationships, what was needed next, and what needed to be found. Grade
46 Examining the Ramp: Looking at Responses of the Early 4 s (2932) The ramps for the seventh grade test: Will it Happen? Part 3 Finding probability of a compound event o Understanding the difference between the number of combinations that create a favorable outcome and the number of ways that combination can be made o Finding sample space for a compound event o Quantifying the probability with a numerical expression Odd Numbers  Part 6 Working backwards from a total to a figure number o Understanding relevant features of the pattern o Making connections between the number being squared and the number of elements in the pattern Pedro s Table  Part 4 Making a mathematical justification o Quantifying the relevant multiples of 5 and the factors of 12 o Showing the connection between multiples of 5 and factors of 12 o Relating the information back to the assertion being proved Winter Hat Part 3 Finding the surface area of the hat o Recognizing all the pieces in the hat and finding the area of each o Being able to see how the trapezoid connects to other parts in the pattern to find the appropriate dimensions (especially the connection between the circumference of the circle and the top of the trapezoid o Being able to identify and use the appropriate dimensions in the trapezoid needed to calculate area o Realizing that there are 8 trapezoids in the diagram o Being able to organize and keep track of information in a longer chain of reasoning Sale! Part 1 Finding equivalent units in order to compare o Fixing either the cost or the quantity to make comparisons between options o Understanding how to compare 3 items with 2 items Grade
47 With a group of colleagues look at student work around points. Use the papers provided or pick some from your own students. How are students performing on the ramp? What things impressed you about their performance? What are skills or ideas they still need to work on? Are students relying on previous arithmetic skills rather than moving up to more grade level strategies? What was missing that you would hope to see from students working at this level? When you read their words, do you have a sense of understanding from them personally or does it sound more like parroting things they ve heard the teacher say? How do you help students at this level step up their performance or see a standard to aim for in explaining their thinking? Are our expectations high enough for these students? For each response, can you think of some way that it could be improved? How do we provide models to help these students see how their work can be improved or what they are striving for? Do you think errors were caused by lack of exposure to ideas or misconceptions? What would a student need to fix or correct their errors? What is missing to make it a topnotch response? What concerns you about their work? What strategies did you see that might be useful to show to the whole class? Grade
48 Student 1 Grade
49 Student 2 Grade
50 Student 2, part 2 Student 3 Grade
51 Student 3, part 2 Student 4 Grade
52 Student 4, part 2 Grade
53 Student 5 Grade
54 Student 5, part 2 Grade
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