Ratios and Proportional Reasoning Assignment #4 Erica Sautter Marist College
7 th Grade Ratios & Proportional Reasoning Sautter 2 Abstract One of a teacher s many responsibilities is to continually assess each of their student s learning throughout each topic or unit studied in class. Using assessments such as traditional tests and quizzes alone is not ideal as they tend to cover only one small portion of the content therefore not truly covering each student s fullest ability. To fill this void, teacher s now include performance based assessments in the classroom. This type of assessment allows the teacher to assess multiple objectives at various levels of Bloom s taxonomy at once, something traditional assessments cannot. This paper will follow a seventh grade mathematics teacher through the process of creating a performance based assessment and a rubric to be used with the assessment for a unit on ratios and proportional reasoning based on the learning objectives shown in a test blue print created for the unit. The teacher will then reflect on why this particular performance based assessment is appropriate for this particular class.
7 th Grade Ratios & Proportional Reasoning Sautter 3 In today s world of education teachers are being asked to create new ways to assess their student s learning. Traditional assessments such as tests and quizzes are no longer filling all of the gaps. Therefore, in order to fill these voids teachers are being asked to incorporate performance based assessments into their curriculum. Performance based assessments are beneficial as they assess students at much higher levels of Bloom s taxonomy than traditional assessments do as well as cover several objectives at one time. When it comes to administering traditional assessments students may know what the correct answer is; however they may not understand why that is the correct answer or be able to explain to a classmate or the teacher how they determined their answer (Park & Lane, 1996). Performance based assessment takes the guessing of how an answer was achieved out of the equation as students are expected to explain how they arrived at their answer. Performance Based Assessment Performance based assessments provide teachers with resourceful information about their students thinking and reasoning as they work through problems to reach solutions and describe in detail explanations for the solutions they reached (Park & Lane, 1996). Performance based assessment works by allowing teachers to observe students as they demonstrate their understanding of multiple skills and knowledge of the subject generally during a hands-on project or task (Hancock, 2007). Applying performance based assessments to the classroom brings along many advantages to both the student and the teacher. One advantage of such an assessment is that it allows the teacher to clarify why students are expected to reach these learning targets (Hancock, 2007). Many times giving students a performance based assessment allows the teacher to connect
7 th Grade Ratios & Proportional Reasoning Sautter 4 learning to a real life situation. Performance based assessments are also beneficial because they allow teachers to assess each student s thought process and the extent of their knowledge of the skills and knowledge being assessed all while they are completing the assessment (Hancock, 2007). This is vital to the teacher as they can draw conclusions as to how students would approach such a problem if they were to face a similar scenario in the real world during their adult life. This is important as one of the goals of school is to prepare our students for the real world and the fact is there are no paper and pencil exams in the real world, but there are performance based assessments. When it comes to performance based assessments the advantages usually outweigh the limitations and disadvantages, however that does not mean they do not exist. Performance based assessments have three primary limitations including: reliability, sampling, and time (McMillan, 2011). One of the biggest disadvantages and limitations of performance based assessments is time. It takes teachers ample amounts of time to create good performance based assessments based upon the standards and objectives being covered in each unit. Then it takes a lot of time for students to actually complete the assessment which then teachers have to take the time to sit down and grade (McMillan, 2011). Another limitation of performance based assessments is reliability. With performance based assessments there is no one answer that is expected for each problem. This allows room for inconsistencies in performance levels of students which can produce inaccurate conclusions over time (McMillan, 2011). This reliability can be lessened with the use of rubrics, which will be discussed a little later, as they provide a guide as to what the teacher is expecting students to show through the assessment. Sampling is another weakness of performance based assessments as there are only a few samples teachers will be able to use with classes due to the time they take to create (McMillan, 2011). With this type of assessment
7 th Grade Ratios & Proportional Reasoning Sautter 5 samples do not provide adequate information to students as these are previous students own interpretation of the assessment situation. The greatest disadvantage of performance based assessments would be the time teachers have to take to create them as well as a rubric to use to grade them, then the time students have to spend actually performing the task and the time that teachers have to spend grading and providing feedback. Although this is the greatest disadvantage with this assessment this is definitely outweighed by all that a performance based assessment has to offer our students as they reveal each students understanding of the content. It is important to remember as we assess these assessments the length of the response is not what is important, it s the extent of the content in the response that matters most (Park & Lane, 1996). Rubrics Rubrics can be used in any content area and are especially useful when it comes to performance based assessments as they allow students to know exactly how they will be graded and that all grading will be done objectively by the teacher (Gezie, Khaja, Chang, Adamek, & Johnsen, 2012). Rubrics take out the idea that an assignment will be graded subjectively based on who the student is and how the teacher feels about the student by providing guidelines for grading. Rubrics are guidelines that clearly articulate performance expectations and proficiency levels (Gezie et al., 2012, p.422). They are generally handed out to students as they begin the assessment allowing students to know what qualities the teacher will be looking for in their work. Rubrics can be used in any content area for many different purposes. They can be used for individual and group projects, oral presentations, practicing favorable behaviors, and any
7 th Grade Ratios & Proportional Reasoning Sautter 6 other type of performance based assessment (Gezie et al, 2012). Rubrics aid in enhancing student learning and their end product by addressing the key elements of the learning goal for the assignment (Gezie et al., 2012). There are two types of rubrics that teachers can use: holistic and analytic. With holistic rubrics the teacher gives the assessment an overall score where components are not scored individually (Gezie et al., 2012). In analytic rubrics the teacher separates the rubric into the different components of the overall product and each component is given an individual score. Then each of the individual components scores are added together to obtain the total score (Gezie et al., 2012). Holistic rubrics are generally used when minor errors in the final product can be tolerated as long as the overall focus and quality of the product remains high. These types of rubrics are quick to make and allow for quick grading as you are only grading on the overall quality of the product, not individual parts (Gezie et al., 2012). Generally, holistic rubrics are used when the nature of the assessment is summative (Gezie et al., 2012). This scoring method can be used on extended response essays where student responses are long and there are multiple responses that could be given (McMillan, 2011). Analytic rubrics take ample amounts of time to create as they are very detailed. Analytic rubrics are used when a particular response is expected in order to assess the knowledge and skills of the student (Gezie et al., 2012). This type of rubric is beneficial when teachers give a project and are expecting particular information to be presented in the end. Because this type of rubric is broken into components of the project it is very detailed in that each category is broken down into scores and under each score there is a short description of what must be done to
7 th Grade Ratios & Proportional Reasoning Sautter 7 receive each of the particular scores. Therefore due to this detail there is a high degree of feedback presented in this type of rubric (Gezie et al., 2012). Although it does take ample time as the teacher must sit down and decide what a student must do in terms of organization, for an example, to earn a score of a 0, 1, 2, or 3. In creating this type of rubric it is helpful to keep in mind that it is better to include few categories where each level is easily differentiable than to have so many categories you cannot differentiate the levels presented (Gezie et al., 2012). Both analytic and holistic rubrics have their place in the classroom when it comes to performance based assessments. However it is up to the teacher to decide what is important and what is being measured in order to know which type of rubric is the best fit for each performance based assessment used in the classroom. Performance Based Task The following performance based assessment will be incorporated into the 7 th grade unit on ratios and proportional reasoning. This performance based assessment will take the concepts and skills taught throughout the unit to the next level as it applies these concepts and skills to real life. Previously I have created a summative assessment in the form of a traditional paper and pencil exam for the concepts covered in this unit. Unfortunately that type of assessment only tells the teacher if the students know the correct answer or not, it does not do a good job explaining to the teacher how the student arrived at a particular answer like performance based assessments do. Therefore for this performance based assessment students will be asked to work with their learning partners as they create an island treasure hunt map (Appendix B) using skills and concepts they previously learned throughout the unit regarding map scale.
7 th Grade Ratios & Proportional Reasoning Sautter 8 Targeted Learning Objectives This particular performance based assessment applies to the following points on the unit s test blueprint (Appendix A) under the content area of proportions: presented. Understand: Students explain how they create proportions based upon relationships Analyze: Students will describe the proportional relationships displayed in tables, graphs, & verbal descriptions. Evaluate: Students will justify how they used proportions to solve problems Create: Students will create their own map scale project using proportions to find model and actual distances. Targeted Common Core State Standards The following Common Core State Standards are addressed through this particular performance based assessment: 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units 7. RP.2 Recognize and represent proportional relationships between quantities. 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7 th Grade Ratios & Proportional Reasoning Sautter 9 Island Treasure Hunt Map Overview The students will work with their learning partners to complete this performance based assessment. I chose to have them work in these pairs because then they are automatically paired up with someone with similar mathematical ability. This is beneficial as my class is very diverse and I want students to show all of what they know about map scale and proportions and how they relate to real life. Each set of partners will work on creating an island treasure hunt map which will allow for student creativity. Each student will be given a packet outlining the guidelines of the project (Appendix B) as well as a grading rubric the first day we work on the project in class. Students will be given adequate time to complete this project during class time, however they will be given suggested guidelines as to what should be finished each day so that they will be on track to finish on time. The first day we will read through the guidelines for part one as a class. This will allow for students to ask questions about items they may not fully understand and it will also allow them to see what is expected of them prior to getting started. One of the first things each group will have to decide on is what scale to use for their map. They will be told that the scale must use centimeters to kilometers and that they may not choose a 1 cm to 1 km or 1 cm to 2 km scale. This scale will have to be realistic and allow for them to include all necessary pieces given the specified guidelines. Each map must be at least thirty kilometers wide and twenty-five kilometers long at its longest parts. How large the island is and the shape it takes on is left up to the creativity of the student. There must be at least three different types of terrain that are a minimum of five kilometers apart as well as four landmarks that are also at least five kilometers apart. Their island
7 th Grade Ratios & Proportional Reasoning Sautter 10 must also include a water source of some type that is at least eight kilometers long. Because they are making an island treasure map they must include an actual treasure that is exactly ten kilometers away from any point of their water source. Completed maps must include a key on the map depicting what the different symbols mean in regards to terrain, landmarks, and treasure as well as include the scale of the map. As students work through creating their maps they will be asked to complete a chart showing all measured and actual distances from various points on their map. The actual distances between any two points will be determined based upon the measured distance on the map and the scale that is associated with their particular map. Students will be asked to show three of the calculations they made in determining these distances to show they know how to set up and solve a proportion using a scale based upon measured distances. The last part of the project will ask students to answer several questions pertaining to their map. A few of these questions will serve as reflection questions as they ask students to explain and describe how they determined some of the different parts of their map and what some of the things they did mean. Students will be given an analytic rubric (Appendix C) that grades them on the following components: use of scale factor, using proportions, using cross products or equal ratios, drawing/model is done to scale, conceptual understanding, use of vocabulary, and presentation/creativity. For each category they can receive a score between one and four. A one would represent they showed multiple misunderstandings in the concept or failed to present it at all. A score of four would represent that they had a strong understanding of the concept with very minimal errors. The total score they may receive on this project is twenty-eight points, therefore
7 th Grade Ratios & Proportional Reasoning Sautter 11 their final grade will be the sum of each individual components awarded points divided by twenty-eight. Reflection I feel that this is a good performance based assessment for this particular unit on proportional reasoning as this represents a real world application to the unit on proportions. Many of my previous assessments for this unit did not focus on the content area of proportions presented in my test blueprint. Instead they focused on using proportions in real world examples associated with applying percents and ratios. Although these concepts use proportions, proportions are used with other content, not by themselves as they are in portions of map scale. This performance based assessment is appropriate for my unit as it covers many of the standards I will be covering throughout the unit. Throughout the unit I will be covering four Common Core State Standards and this performance based assessment will cover parts of three of them. Standard 7.RP.1 wants students to compute unit rates associated with ratios of lengths. Students show their understanding of unit rates when they determine the scale for their map. They then use this scale, which is a ratio in itself, to determine the lengths of distances on their map. One way students could compute their unit rates would be to draw an island on their poster paper and then develop a scale that would be appropriate based upon the requirements of the project. Standard 7.RP.2 asks students to recognize and represent proportional relationships between quantities. Students are asked to show and discuss the proportional relationships they see in their map both in the constructed response questions in their packet as well as when they solve to find the measures of each of the distances between points.
7 th Grade Ratios & Proportional Reasoning Sautter 12 Students meet standard 7.G.1 during most of the construction of the map and while completing the chart on page two of the guidelines packet. This standard asks students to compute actual lengths as they solve problems that involve scale drawings. Since this project is a scale drawing and they are computing these distances and lengths they are addressing this standards. This assignment will help students learn the content presented about proportions by allowing them to apply this knowledge to a hands on task. Creating this type of visual will assist the visual learners in deepening their understanding of the concepts presented. Many students may have a difficult time understanding what it means to have a scale and how one would actually go about computing a scale and using it to find useful information. It is my hope that by allowing students to be creative in their creation they truly deepen their understanding about the importance of proportions and why they must learn about them for reasons other than the state said we had to. I feel that this performance based assessment is also appropriate because it taps into the students thinking processes as they are asked to describe, justify, and explain why they did different things for different parts of the assignment. An analytic rubric is beneficial for this project because there are many different parts that I as the teacher want to assess. Additionally, there is not a broad range of responses I am looking for as many of their answers on the extended response questions should focus around the same ideas. Overall, this is a good performance based assessment for my seventh grade students as they learn about ratios and proportional reasoning as it allows them to use their creativity while applying the skills and knowledge they have learned about ratios and map scale to real world
7 th Grade Ratios & Proportional Reasoning Sautter 13 concepts as they create a fun map. This assessment also shows me, the teacher, if my students truly understand the meaning of proportions and their use or if they only know how to find one, not truly understanding them.
References Gezie, A., Khaja, K., Chang, V., Adamek, M. E., & Johnsen, M. (2012). Rubrics as a Tool for Learning and Assessment: What Do Baccalaureate Students Think?. Journal Of Teaching In Social Work, 32(4), 421-437. Hancock, D. R. (2007). Effects of Performance Assessment on the Achievement and Motivation of Graduate Students. Active Learning In Higher Education, 8(3), 219-231. McMillan, H., James. (2011). Classroom Assessment: Principles and Practice for Effective Standards-Based Instruction. Boston, MA: Allyn & Bacon. Parke, C. S., & Lane, S. (1996). Learning from performance assessments in math. Educational Leadership, 54(4), 26-29
7 th Grade Ratios & Proportional Reasoning Sautter 15 Appendix A Ratios and Proportions Test Blueprint Proportional Reasoning Unit Ratios Remember Understand Apply Analyze Evaluate Create Totals Students will define what it means to have a ratio. 3/ 3% Students will explain the purpose of a rate. 2/ 2% Students will write real world ratios. 5/ 5% Given a table, graph, verbal description, or chart students will identify the rate shown. Students will evaluate ratio/rate problems. 10/ 10% Students will create a ratio/rate problem of their own to solve. 5/ 5% 33 / 33% Proportions Students will describe the similarities between creating a ratio and creating a proportion. Students explain how they create proportions based upon relationships presented. Students will write real world problems that require proportions to be solved. 8/ 8% Students will describe the proportional relationships displayed in tables, graphs, & verbal descriptions. Students will justify how they used proportions to solve problems. Students will create their own map scale project using proportions to find model and actual distances. 39/ 39% 5/ 5% 3/ 3% 5/ 5% 10/ 10% 4/ 4% 12/ 12% Applying Percent Students will recall how to convert between fractions, decimals, and percents. 3/ 3% Students will explain their thought processes when finding percentages with or without a proportion. 5/ 5% Students will demonstrate how they think about using percents in real life in relation to tax, tip, discounts, interest, etc. 3/ 3% Students will use given information to solve for percentages and other missing information. 10/ 10% Students will assess when it is appropriate to use a proportion and when they should use the percent equation to solve problems. Students will create their own situations to determine items such as tax, tip, discounts, interest, commission, etc. 5/ 5% 28/ 28% 2/ 2% Totals 11/ 11% 10/ 10% 13/ 13% 28/ 28% 16/ 16% 32/ 32% 100/ 100%
7 th Grade Ratios & Proportional Reasoning Sautter 16 Appendix B Island Treasure Hunt Due: Goal: Create an island using a scale of cm = km **You may NOT use a scale of 1cm:1km or 1cm:2km** Part 1: o Name your island. o Include a key with landmarks, terrains, water sources, the treasure, and a scale. o The island must be at least 30 km wide and 25 km long. o Include three types of terrain (mountains, jungle, quicksand, etc.) that are at least five kilometers apart. o Include four landmarks (graveyard, a tree, a monument, etc.) that are at least five o kilometers apart. o Include a water source that is at least eight kilometers long. o Include an X where the treasure is located exactly ten kilometers away from a water source. o Name and label all landmarks, terrains, water sources, and treasures. Part 2: Fill out the table to calculate the distances between various landmarks on your map. Round to the nearest tenth. Answer the questions following the table. Be sure to include vocabulary terms! Keep your work neat and organized. Review the rubric for how your island will be graded. Be CREATIVE and HAVE FUN!!!
7 th Grade Ratios & Proportional Reasoning Sautter 17 Fill in the table below. Show both measured and actual distances in each box. Use the next page to show your work on how you found the selected distances. Distances Landmark 1 Landmark 2 Landmark 3 Landmark 4 Terrain 1 Terrain 2 Terrain 3 Water Source Treasure Landmark 1 Landmark 2 Landmark 3 Landmark 4 Terrain 1 Terrain 2 Terrain 3 Water Source Treasure What scale did you use? What was your scale factor?
7 th Grade Ratios & Proportional Reasoning Sautter 18 Display your work for the following three table distances. Be sure to explain where the numbers came from for each scenario. Landmark 1 to Landmark 2: Landmark 2 to Landmark 3: Landmark 3 to Landmark 4:
7 th Grade Ratios & Proportional Reasoning Sautter 19 Question Sheet 1. Explain how you created the scale you used for your island based upon the requirements? 2. Explain your strategy of setting up and solving the proportions on the previous page. Make sure you include the words scale, cross products, units, and actual distance in your response. 3. Describe the proportional relationship presented in your map. Why is this beneficial to someone reading your map?
7 th Grade Ratios & Proportional Reasoning Sautter 20 4. On your map draw a path that will lead you to the treasure. This path should start from the landmark furthest away from the treasure. The path you take must stop at every landmark on your map before reaching the treasure. On the lines below, write out how you would tell a friend to travel from the first landmark to the treasure using directions and actual distances. For example: If you start at the monument you must walk 4.6 miles northeast to reach the apple tree.
7 th Grade Ratios & Proportional Reasoning Sautter 21 Appendix C Name: Rubric for Island Treasure Hunt Scale Drawing Project Total /28 Use of Scale Factor Using Proportions Using Cross Products or Equal Ratios Drawing/Model is Done to Scale Conceptual Understanding Use of Vocabulary Drawing Presentation and Creativity Scale factor is used correctly for all calculations. Correctly set up proportions for scale drawings. 4 3 2 1 Demonstrates the ability to use cross products or equal ratios accurately to solve proportions. Measurements are correctly rounded to the nearest tenth with 0 to 1 calculation errors. An appropriate scale was used for the drawing. All landmarks, terrains, water sources, and treasure were included. Model is drawn accurately with 0 to 1 minor errors. All questions were answered fully and completely showing a strong understanding of the concepts. Strong details. Answers to questions included appropriate math vocabulary. Finished product is creative, neat, organized, colored, and includes all required information. Scale factor is used correctly for some calculations. 1-2 mistakes. Proportions are set up correctly for some calculations. 1-2 mistakes. Uses cross products or equal ratios to solve proportions. Shows all work and rounds accurately to the nearest tenth. 1 to 2 calculation errors. An appropriate scale was used for the drawing. Shows at least 8 of the required landmarks. Model is drawn accurately with 2-3 errors. All questions were answered showing a good understanding of the concepts. Has 1 mis-understanding. Some details. Answers to questions included some math vocabulary with minor errors. Using wrong term or definition Finished product is creative, neat, colored, organized, and is missing 0-2 required pieces of information. Scale factor is used correctly for some calculations. 3-5 mistakes Set up proportions that are incorrect for scaled drawings. Uses cross products or equal ratios to solve proportions. Shows all work and attempts to round. 3 to 5 calculation errors. An appropriate scale was used for the drawing. Shows at least 7 of the required landmarks. Model is drawn accurately with 4-5 errors. All questions were answered showing some understanding of the concepts but no details were included. Answers to questions included minimal math vocabulary or contains major errors. Lacks organization, missing 2-4 pieces of information. Not colored. Scale factor is used correctly for some calculations. 5+ mistakes Fails to use proportions to show change of scale. Fails to use cross products or equal ratios to solve proportions. Does not attempt to round answers. Shows no work or has missing work. More than 5 calculation errors. Fails to use an appropriate scale for the drawing. Shows fewer than 6 of the required landmarks. Model has more than 5 errors. Fails to show any understanding of the concepts. More than 3 misunderstandings. Fails to include any vocabulary in written responses. Fails to show any organization, is not colored, and is missing more than 4 pieces of information.