Facilitator’s Guide To Leading the Scoring Session Level 2

Facilitator’s Guide

To Leading the Scoring

Session Level 2 An Introduction for High School

Math Teachers

This packet contains the following:Facilitator's Guide to Leading the Scoring Session Student work with commentaries

Oregon Department of Education 2011 12 Office of Assessment and Information Services

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intentionally left blank.

Intro to Math Scoring Guide – Level 2 Training Facilitator’s Guide to Leading Scoring Session: High School Math Teachers

Suggestions for Use of Student Papers

Explain that this part of the training will focus on applying the math scoring guide to student work, with an emphasis on the differences between the 3 and 4 score points for all dimensions and whether or not the student work meets the essential skill requirement for an Oregon diploma. Distinguishing between the 3 and 4 scores points is important for three main reasons:

1) the 3/4 call is the most critical one for students because it determines whether or not they earn passing scores;

2) it is most likely the decision that will have to be made most frequently--most papers fall into this category;

3) it is relatively easy to identify papers that both exceed the standard and those that fall far below the standard.

It isn’t worth the investment of limited time to debate the 5 versus 6 or the 1 versus 2 score points. An additional concern at the high school level is whether or not a passing work sample meets the essential skill requirement for graduation. In order to qualify, the student work must reflect high school level standards

Farmer John Mathematics Content Standard Assessed by This Task: H.1G.5

Language From Achievement Level Descriptors for Meets which Describes the Required Skills: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context- based problems.

Work the Task (slides # 24 and 25) Work the first task (Farmer John) independently, then pair share solutions - followed by a group discussion –including examples of a variety of approaches (preferably provided by participants), possible student approaches, and the key concepts indicated by the task.

Anticipated Student Strategies: Guess and Check, Table, Writing a Quadratic and using either factoring (in which case, this task would also include standard H.1A.5 and H.3A.5), graphing or the quadratic formula to solve.

Scoring of Student Papers To prepare for the discussions that follow, the facilitator should read the paper commentaries included as a separate document and make notes on their copies of the student papers. A separate single sheet is also included with all of the scores for the training papers for both tasks.

Close Reading: Scoring Guide – Proficient - vs - Not quite Proficient • Participants review scoring guide 4 for Each Dimension

• They should identify words and phrases that distinguish a four for each dimension

• Facilitator then clarifies the factors that usually differentiate a 3 from a 4 in each dimension

Farmer John Student work (slide #26) Tell Participants that the first paper they will be discussing has been determined to be “adequate” in all dimensions – in other words that it is a “passing” work sample. Inform them that they will be reading it with this is mind and that the discussion will focus on finding language from the scoring guide that backs up the scores of 4 for each dimension

Paper 1 Farmer John TR 1 Participants Read Farmer John (TR 1), Facilitator leads participants through a discussion of why this paper earned scores of 4 in each dimension – using commentaries sheet, and notes as well as input and questions from participants. Focus on having participants find words and phrases in the scoring guide that fit the student work – if participants express concerns about these scores, indicate that these scores are the result of multiple reviews by multiple expert scorers – and that these are meant to help them become calibrated to the scoring guide, and move on.

Paper 2 Farmer John TR 2 Before looking at this paper, have participants review the language in the first 2 dimensions - MS, and RS and discuss how these dimensions are similar and how they differ,

Facilitator asks participants to read through Farmer John TR 2 and record tentative scores for the first 2 dimensions on their papers including words or phrases from the scoring guide that led to their scores. When participants seem to have completed this task, explain to them that they will now have the opportunity to discuss and defend their scores. Emphasize that this is meant as an exercise in “calibration” and that no one is expected to be perfect, but that the discussion is an important way to work toward “proficiency” as scorers.

Ask participants to show fingers to indicate what score they gave this paper for Making Sense of the Task . Give the group an “overview” of the scores given, point out that scorers that are within one point of each other are very close and hopefully the discussion of their scores will help clarify their thinking. Next reveal that the expert scorers gave this student a score of 4 in this dimension and explain why, using the commentary to assist you. Encourage questions/discussion including asking participants what words or phrases from the scoring guide led them to their scores– but be sure to move on after a reasonable amount of discussion. Point out that many examples of a variety of student work pieces need to be seen to become a good rater.

Repeat the above process for this paper for Representing and Solving the Task, if there are any participants with 2 or 6 fingers up, point out that these scores need to be calibrated and ask if any of the people with these scores would like to share why they thought the paper earned this score – then lead the discussion to help them move to comfort with the given scores.

Next – review the descriptions of CR, AC, and RE dimensions and ask participants to identify key words in each, then have the participants go back and give tentative scores for each of those dimensions for Farmer John Paper #2 and repeat the sharing process for each of these.

After all dimensions have been scored and discussed, point out that this is not a proficient work sample, but that it is very close, and hopefully this discussion has helped them to begin to make the differentiation between 3’s and 4’s and to separate the dimensions. (Slide #28)

NOTE:

This task includes scores of 3’s and 4’s which means it is NOT a passing work sample – but it is close. This offers a good time to discuss the fact that students may be given an opportunity to rework a work sample – if the teacher feels that the student is close enough to passing to be able to make changes that could make it “proficient” without any reteaching or coaching (this may also be a good time to mention the official “feedback form” and the information about reworking opportunities in the Test Administration Manual. There is an example to use at the end of the Farmer John scoring.

A Brief Look at a High and Low Paper and a Few examples of 6’s (papers TR 3, TR 4, and TR 5): Even though the most critical call is between the 4 and the 3 score points, it is important to recognize the high and low papers. Participants will just read and briefly discuss one of each as well as a paper that was included so that participants could have the opportunity to see examples of scores at a level of 6 and appreciate the fact that 6’s are truly very special. Manage discussion based on your perception of participant’s needs and time available. (Slide #29)

Using the Feedback Form: In this scenario, a student has solved the Farmer John task (work samples may be found in the Level 2 training materials) by using a “guess and check” method to solve the task accurately. This may generate discussion among your participants about the use of this strategy to demonstrate proficiency in Essential Skills Apply Mathematics. Others may defend this student’s method by claiming the high school mathematics standard targeted does not stipulate what strategy to use.

If you believe the student needs to revise this work, then use the feedback form to communicate this to the student. Note: This would be a good opportunity to ask your audience how they might fill out the feedback form to communicate this. Then reveal how this particular one was filled out.

Next, look at how the student revised his/her work. Notice how the standards and achievement level descriptor are written at the top of the page. Notice how the directions include the sentence: Use equations, table, and/or graphs to solve this problem.

This collection also illustrates the value of revision, provides the confidence for the level of mathematics and the “difficulty” of the task.

Some of your participants may ask: Are we making a rule that if a student uses guess and check, it’s not ok at high school? Is that an official position of ODE?

Response from Derek Brown, Manager of Essential Skills: I wouldn’t say there is an “official” position, but would point out that work samples are one acceptable sourceof evidence that a student has demonstrated proficiency in the Essential Skills. If a student has used the “guess and check” method to complete a work sample, how confident would the rater/teacher/district/school board be that the student has acquired the knowledge and skills certified by awarding a diploma?

That’s what this issue can really be boiled down to: does the rater feel confident that the work sample(s) are reflective of independent student work and are sufficient evidence to support the student has demonstrated proficiency in the Essential Skill? If so, and the other graduation requirements have been completed, the student should be awarded a diploma to certify the knowledge and skills they’ve acquired/demonstrated.

I’m not a math content person, but imagine there may be situations where “guess and check” are acceptable steps in the problem solving process. If so, and the student has used the method in an appropriate manner (as part of a systematic process) and can justify their thinking, I would tend to think it’s acceptable. In cases where the student is clearly guessing (correctly) but not basing it on any clear understanding of content, I’d be less confident in making a diploma decision.

If I were in a local district, I’d make a strong push for developing a local performance assessment policy that clearly describes how these and other issues will be managed.

Response from Jim Leigh, Office of Assessment and Information Services: As we discussed this fall, the task instructions could include “Use equations, tables, and/or graphs to solve this problem.” If the rules are not spelled out in the prompt, then I feel that any method the student uses should be accepted. If they can just look at the problem and guess the answer, then it may not be a high school-level task. Bike Rental Repeat the process of working the task, pair share, and group discussion of this task just like Farmer John (Slides 30 and 31)

High school content standard: H.3A.5

Achievement Level Descriptor: Determine the vertex of a quadratic function graphically and/or algebraically.

Anticipated Student Strategies: Table, graphing, writing a quadratic equation and using it to find the vertex and interpret what it means

Paper TR6 Have participants score the first bike rental paper (TR 6) and record their scores and key words and phrases in all 5 dimensions

Lead a whole group discussion of the scores and commentaries after revealing that this paper was proficient in every dimension (Slide 32)

Have participants individually score paper TR 7 one dimension at a time, then pair share and have these pairs agree on scores before revealing the anchor scores to the whole group – lead discussion using information from commentary.

Repeat scoring and pair/share (or consensus for small group) with each of papers TR 8, TR 9, and TR 10 possibly only asking for discussion of certain dimensions for each, depending on time and needs of the group, follow each with a whole group discussion of anchor (expert) scores.

Option: If there is time, raters could go back and score any paper for dimensions not previously scored.

Depending on the size and make-up of your audience, possibly give a copy of the anchor score to “table leaders” and have them do the reveal of the scores, and calibration discussion in these smaller groups.

Training Papers Math Level 2 for High School Math Teachers

Paper # MS RS CR AC RE

TR 1 4 4 4 4 4

TR 2 4 4 4 3 3

TR 3 3 3 2 3 1

TR 4 5 5 5 5 5

TR 5 5 5 5 5 6

Revision Papers

R 1 4 4 4 4 4

R 2 6 6 5 6 6

TR 6 4 4 4 4 4

TR 7 4 4 4 4 4

TR 8 4 4 3 3 2

TR 9 4 3 4 3 2

TR 10 5 6 6 5 6

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Participant Score Recording Sheet Paper

# Title MS RS CR AC RE

TR 1 Farmer John

TR 2 Farmer John

TR 3 Farmer John

TR 4 Farmer John

Space is provided in this table to allow you to record your original score, the expert score and any comments you wish.

Paper# Title MS RS CR AC RE

TR 5 Farmer John

TR 6 BikeRental

TR 7 BikeRental

TR 8 BikeRental

TR 9 BikeRental

TR 10 BikeRental

Space is provided in this table to allow you to record your original score, the expert score and any comments you wish.

Mathematics Work Sample High School 2011 – 2012 – Farmer John

Use the information provided to solve the problem listed below. Be sure to show your work at all phases of problem-solving. Refer to the Student Problem Solving Tips to receive the highest score in each of the five areas. Algebra Geometry Statistics Content Standard: CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.A.REI.4 Solve quadratic equations in one variable. Achievement level descriptor: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context- based problems. Name: ______________________________________

School: _____________________________________

Teacher: ____________________________________

Farmer John has a rectangular holding pen that measures 10 yards long and 5 yards wide to contain his cattle. He is acquiring more cattle from the neighboring famer and wants to add the same amount of fending to each side to create a new holding pen that encloses 176 square yards. How much should Farmer John add on to each side of his existing holding pen to achieve his goal?

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Mathematics: Essential Skills Scores and Commentary

Work Sample Title: Farmer John Paper Number: TR1 Algebra Geometry

Statistics High School Content Standard: H.1G.5: Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions. CC.9-12.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.A.REI.4: Solve quadratic equations in one variable.

Achievement Level Descriptor: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context- based problems.

MS RS CR AC RE 4 4 4 4 4

Making Sense of the Problem:The interpretation of the task is adequately developed and displayed. The student adequately interpreted the task into finding an additional length of fencing to add to both dimensions in order to reach he stated area. The student displays his/her understanding through the accurate display of finding area (diagram at the top of the paper) and the chart, which clearly shows adding the same amount to each dimension and searching for the target area of 176 square yards.

Representing and Solving the Problem:The representation of the guess and check strategy is both effective and complete. The student demonstrates a systematic way to pinpoint the correct number of additional yards by making an initial guess and adjusting up or down until they reach 176 square yards.

Communicating Reason:The completeness of the guess and check table, especially the check column makes the work clear and coherent. It leads to a clearly identified solution.

Accuracy:The solution given is correct and mathematically justified and supported by the work.

Reflecting and Evaluating:The solution is stated within the context of the problem. The review of the concepts and the reasonableness is embedded in the table. The solution is also justified once more by recalculating the side lengths and area. It is not a 5 or 6 because the student does not re-work the task using a different method or show evidence they’ve considered other outcomes or interpretations.

TR2






Making Sense of the Problem:The student has adequately interpreted and displayed the concepts of area and dimension as well as the need to add an equal length to each dimension

Representing and Solving the Problem:Both the scale drawings and the systematic table are effective and complete but represent basically the same approach. The student comments that the table is more effective than measuring one more yard on each side but this is not enough for a score of 5 or 6.

Communicating Reason:The communication of the reasoning follows a clear and coherent path illustrated by the drawings, the table and the associated commentary describing the thought process. The work leads toward a correct solution. Although the student does not use precise mathematics vocabulary (“long side numbers” and “wide side numbers” instead of length and width), this does not create a gap for the reader.

Accuracy: The student circled the last row in the chart, but does not answer the question in the originalproblem. As a result, the work is partially complete.

Reflecting and Evaluating:The solution is not stated in the context of the problem. Both approaches lead to the same result, which would be correct if a solution had been stated.

TR3


Work Sample Title: Farmer John Paper Number: TR3 Algebra Geometry Statistics

High School Content Standard: H.1G.5: Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions. CC.9-12.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.A.REI.4: Solve quadratic equations in one variable.


MS RS CR AC RE3 3 2 3 1

Making Sense of the Problem:The interpretation is partially displayed. The student does not reveal where the dimension of 16 comes from.

Representing and Solving the Problem:The strategy displayed is partially complete. The student uses trial and error but doesn't show any results of the trials until the final result.

Communicating Reason:There are significant gaps in the communication of the reasoning, making it underdeveloped. In addition, the answer provided doesn't make much sense, even though the student found the correct dimensions.

Accuracy:The student has the correct dimensions but has interpreted them into only a partially correct answer.

Reflecting and Evaluating:Normally, there is some reflection evident in guess and check. Due to partial display of the work, it is not evident here.






Making Sense of the Problem:The student adequately interprets the task into finding an additional length of fencing to ad to both dimensions in order to reach the stated area. The use of quadratic equations shows a more thoroughly developed interpretation, as the student as able to represent the mathematics algebraically. In addition, the student shows two related but different interpretations (using ‘x’ to represent the total additional length of one side or using ‘x’ to represent half of the additional distance as shown in the diagram on page 2).

Representing and Solving the Problem:By using an algebraic approach, the student has generalized the problem. The work is enhanced by the second strategy, adding 3 on each end instead of 6 on one, and the use of a slightly different representation (system of equations). The student expresses the solutions in set notation.

Communicating Reason:Although unnecessarily wordy at times, the commentary allows the reader to move easily from one thought to another. The communication is insightful on the second page. Both approaches are enhanced by graphics. Appropriate mathematical language is used throughout.

Accuracy:The answer of 3 yards is not what is generally expected. However, the solution is correct based on the student’s interpretation of the task and mathematically justified and supported by the work. Generalized through the quadratic equations and connected by the graphics, the solution is enhanced. The student does ask the question “I wonder what the dimensions....”but does not go anywhere with this question so it does not contribute to their score.

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Making Sense of the Problem:The student has worked the problem both backwards and forwards, making the interpretation enhanced. The second solution is thoroughly developed.

The student demonstrates his/her interpretation and translation of the task at two different levels. First, the student uses factor pairs in the chart to look for a pair with the same difference from the original dimensions. Next, they show an algebraic interpretation including commentary about why -21 does not fit within this context. Comments like “Next I need to find the value of x, or, in other words, the number of yards that is dded to each side...” make it clear that the student can translate the words in the given task into appropriate mathematics. All of these reasons help make this paper thorough and enhanced.

Representing and Solving the Problem:The representation is enhanced by the two distinct and thorough strategies. The guess and check strategy using factor pairs provides a complete solution and the algebraic solution demonstrates a generalized approach.

Communicating Reasoning:The communication of the reasoning is enhanced by the graphics, labels, and by the commentary, although a little wordy. Although a math teacher would not need the extra commentary on page two (explaining all the steps for the algebraic approach), the student also does a thorough job of connecting all pieces of their work (i.e., “This time I will solve the problem algebraically. I will use...”), allowing the reader to move easily from one thought to another.

Accuracy:The solution is correct and enhanced by the generalized nature of the second approach.

Instructions for Farmer John Example of Revision Materials These materials were assembled to show participants what might happen when a student successfully completes a math problem solving task, but does not use high school level math to do so. The student may be given an opportunity to revise the task if the teacher believes the student could complete the task using high school strategies. Take participants through these materials page by page, explaining how the revision process worked.

1. Pages 1-2 contain the original student work for the Farmer John task. The student uses a guess and check method and illustrations to solve the problem.

2. Page 3 contains the scores and commentary this paper originally received. Note

that the paper scored 4 in all dimensions, but should not be used as a demonstration of proficiency in the Essential Skill of Apply Mathematics because of the lack of high school level math.

3. Page 4 is the Official Math Problem Solving Work Sample Feedback Form that was returned to the student with the original work and possibly a highlighted copy of the scoring guide. The teacher may have also stated that the problem is the lack of demonstration using high school math – but no other instruction or coaching was given.

4. Pages 5-6 show the student’s second version of solving the problem using appropriate mathematical strategies.

5. Page 7 shows the revised Official Scoring Form with the new scores recorded at the bottom of the page.

6. Page 8 is the second version of the scores and commentary so that participants can see exactly how the rater interpreted the student work.

R 1

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Work Sample Title: Farmer John Paper Number: R1 Algebra Geometry Statistics

High School Content Standard: H.1G.5 Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions.


Making Sense of the Problem:

The interpretation of the task into increasing the dimensions by the same amount until reaching the desired area was adequate and complete. The mathematics is not connected or extended to other mathematical ideas.

Representing and Solving the Problem:

The strategy of making a systematic list adding 1 square yard to each original dimension and checking the area until reaching 176 was effective and complete. The strategy is not complex nor is the student able to generalize the situation.

Communicating Reason:

The student clearly represents the two rectangular pens and labels their dimensions. Although the student does not explain what he/she is specifically doing with the table, it is clear that he/she is adding 1 unit to each dimension and then multiplying to find area. However, using a systematic list or guess and check table are not considered elegant or insightful at high school level.

Accuracy:

The solution of adding 6 yards to each side is correct and supported by the work. The work does not include extensions, connections, or generalizations.

Reflecting and Evaluating:

The student summarizes the answer in a sentence at the bottom of the page, indicating they have answered the question asked and did not just give the dimensions. The review is complete, since the student basically re-did the problem again on the second page (adding an additional set of numbers), although they did not restate the solution of adding 6 yards to each side.

**Note: This is a good example of a case where a student earns a "4" in all dimensions for problem solving but does not demonstrate proficiency for Essential Skills.

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G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure tosatisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

R 2

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R 2

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Work Sample Title: Farmer John Revised Paper Number: R2 Algebra Geometry Statistics

High School Content Standard: H.1G.5 Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions. G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Achievement Level Descriptor: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context-based problems.


Making Sense of the Problem:The interpretation of the task into generalizing the amount being added to both dimensions as “x”, and using a quadratic equation to represent the situation is enhanced. The student thoroughly develops the meaning of the x verbally and in the diagram and also looks at the mathematics thoroughly (he/she gets two answers: -21 and 6 but comments why 6 is the only one that makes sense). He/she also describes how to use a graphing calculator to solve the problem, indicating that the student has made sense of this problem in several ways.

Representing and Solving the Problem:The strategy of assigning “x” to the additional amount needed and writing, solving, and interpreting the solution for a quadratic equation makes this paper complex. The additional information on the back about how to use the graphing calculator to solve the problem adds elegance and insightfulness, and connects the problem to another representation.

Communicating Reason:The work follows a clear and coherent path and the student uses mathematical language precisely (I will solve this equation algebraically, Let y1 represent the product of the, set the window so the domain….). The reader is able to move easily from one thought to another through the use of words, symbols and the diagram to the right. The diagram of the two pens helps explain the origin of the equation (10 + x)(5 + x) = 176 but clearly shows that there would be “x” added to both sides of the pen according to the way it is labeled. There is a slight gap between the diagram and the equation (why is it 5 + x instead of 5 + 2x?). The student also does a good job describing the process used with the graphing calculator, but a sketch of the graph and more details in the description would have helped this paper become a 6 in this dimension

Accuracy:The solution of adding 6 yards to each side is correct and justified. It is also extended through the use of technology, since the student uses the technology to check if the intersection is truly the solution they arrived at using the system. The student also thought about the two solutions obtained from solving the system and commented that “6 is the only answer that makes sense”, although they do not say why.

Reflecting and Evaluating: The solution is clearly stated within the context of the task. He/she reworks the task using a graphing calculator (and if you include the original method of making a systematic list, the student has worked the problem 3 ways). He/she also considers the solution of -21 and dismisses this as possible. The approach used with the graphing calculator shows that the student reflected on his/her solution and realized that if 6 is the solution, then it should be one coordinate of the intersection point of the two lines. This evidence moves the score for this dimension to a 6.

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Mathematics Work SampleHigh School 2011 – 2012 – Bike Rental

Use the information provided to solve the problem listed below. Be sure to show your work at all phases of problem-solving. Refer to the Student Problem Solving Tips to receive the highest score in each of the five areas.

Algebra Geometry Statistics

Content Standard:H.3A.5 Given a quadratic equation of the form x2 + bx + c = 0 with integral roots, determine and interpret the roots, the vertex of the parabola that is the graph of y = x2 + bx + c, and an equation of its axis of symmetry graphically and algebraically.CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities.

Achievement level descriptor:

Name: ______________________________________

School: _____________________________________

Teacher: ____________________________________

A local bicycle rental company charges $12 to rent a bicycle. They normally have 300 rentals per month. The company owner has determined that each increase in price of $2 will decrease the number of rentals by 15. What price will maximize the revenue?

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Work Sample Title: Bike Rental Paper Number: TR6 Algebra Geometry Statistics

High School Content Standard: CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities.

Achievement Level Descriptor:


Making Sense of the Problem:Increase in cost related to decrease in rentals is adequately translated, developed and displayed in the chart and displayed again in the graph. The student has clearly interpreted the task and the results.

Representing and Solving the Problem:The strategy of a systematic list is both effective and complete. The graph, relating price to revenue, supports the effectiveness of the approach by indicating the maximum.

Communicating Reason:The chart, the graph, and the verbiage lead the reader to a clearly identified solution with a minor gap. The dependent variable of income on the graph is incorrectly labeled as the number of rentals. The rest of the communication on the page is strong enough that the error is not significant.

Accuracy:The solution is correct and justified by the comparison, in the chart, to the other rental revenues. It is somewhat enhanced by the graph but not enough to warrant a score of 5.

Reflecting and Evaluating:The solution is stated in the context of the task and the use of the values from the chart in the graph indicates reflection beyond what is embedded in the guess and check process.

TR7






Making Sense of the Problem:The translation and interpretation of the concepts (increase in price corresponding to decrease in rentals) is adequately developed by showing the revenue at different prices and showing the relative change in revenue due to each increase, recognizing the effect of a negative change.

Representing and Solving the Problem:The guess and check approach is effective. The student initially finds the revenue resulting from each price change as well as the relative difference in revenues, then concentrates only on the relative change after $18 per rental, applying the effect of a negative change to find the maximum.

Communicating Reason:All steps leading to the solution are clearly shown and explained, implying that the student found the revenues for costs of 20, 22, 24, and 28 but didn’t record them, recording, instead, how much the revenue had changed with each successive price change.

Accuracy:The solution is correct, mathematically justified and supported by the work.

Reflecting and Evaluating:In addition to the embedded reflection throughout the guess and check process, the analysis of the increases provides a check of the solution’s reasonableness.

TR8






Making Sense of the Problem:The concepts of rate increase and corresponding decrease in rentals are adequately developed and displayed. The student shows that an increase of $6 in price will result in a decrease of 45 rentals. The student finds the maximum but doesn’t check.

Representing and Solving the Problem:The guess and check approach is adequate. The student considers increases of $6 and $4 to arrive at the solution more quickly than by increasing just $2 at a time. The student does not, however, consider the revenue at $24 per rental, making this a weak 4.

Communicating Reason:The communication of the reasoning is a little difficult to follow and even though the solution is identified with a box and an arrow, it is not answering the question.

Accuracy:Although the work leads to a correct solution - that solution is not identified. The student has not checked the revenue for a price of $24, causing the result, especially in a guess and check process, to not be adequately justified by the mathematics. There is also an error in recording that 18 X 255 is 4560, instead of 4590.

Reflecting and Evaluating:The only reflection seems to be embedded in the guess and check process and carried out incompletely, failing to check the reasonableness of the solution. The answer is not stated within the context of the task. No reflection beyond guess and check is evident.

TR9






Making Sense of the Problem:The interpretation of the task is adequately developed and displayed. Even though a subtraction error has thrown off the answer, the student has adequately interpreted the key concepts of the task.

Representing and Solving the Problem:Due to the error in the work (for an increase in price from $16 to $18, the student has decreased the rentals by only five) the strategy is only partially effective.

Communicating Reason:The work follows a coherent path and it is easy to tell what the student is thinking. The student uses few words yet has labeled the columns of the table appropriately and the mathematics used is clear and leads to an identified answer.

Accuracy:The solution is incorrect due to a minor error in arithmetic.

Reflecting and Evaluating:A partial review is embedded in the guess and check process. There is no evidence that the student has reviewed the calculations, making the review underdeveloped and ineffective.

TR10






Making Sense of the Problem:This is a thoroughly developed interpretation and translation of the task. The student has identified the variable and labeled the parts of the function, recognizing it as a quadratic relationship. A possible extension (an increase of $3 causing a reduction of 18 rentals) is alluded to but not developed.

Representing and Solving the Problem:All bullets of the 6/5 score descriptors are addressed. The work is insightful in considering approaches that are more effective, complex in the use of a quadratic function, and enhanced by considering multiple ways to arrive at the solution.

Communicating Reason:The student explains the thinking and strategies throughout. Although the flow is sometimes not seamless, it is more than made up for with the use of mathematical language, examples, graphics and explanation of the thinking. The reader is able to move easily through the student’s narrative.

Accuracy:The solution is correct and enhanced by connections to other mathematical ideas. The function and mention of an extension generalize the solution. The solution is mathematically justified 4 times!

Reflecting and Evaluating:Justifying the solution completely, the student has reworked the task using two distinct approaches and found multiple ways to solve the quadratic equation, thereby evaluating the efficiency of each approach. The student has addressed every bullet in the 6/5 column of the scoring guide.

**This paper is an example of student work that fulfills the essential skill requirement for a diploma

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Documents

Facilitator’s Guide To Leading the Scoring Session Level 2