GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS

Save this PDF as:

Size: px
Start display at page:

Transcription

1 GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS UNIT OVERVIEW This 4-5 week unit focuses on developing an understanding of ratio concepts and using ratio reasoning to solve problems. TASK DETAILS Task Name: Ratios and Proportional Relationship Grade: 6 Subject: Mathematics Depth of Knowledge: 2 Task Description: This sequence of tasks asks students to demonstrate and effectively communicate their mathematical understanding of ratios and proportional relationships. Their strategies and executions should meet the content, thinking processes and qualitative demands of the tasks. Standards: 6RP.1 Understand the concept of ratio and use ratio language to describe a ratio relationship between two quantities. 6RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. 6RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, i.e., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. 6.RP.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. 6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Standards for Mathematical Practice: MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. 1

2 TABLE OF CONTENTS The task and instructional supports in the following pages are designed to help educators understand and implement tasks that are embedded in Common Core-aligned curricula. While the focus for the Instructional Expectations is on engaging students in Common Core-aligned culminating tasks, it is imperative that the tasks are embedded in units of study that are also aligned to the new standards. Rather than asking teachers to introduce a task into the semester without context, this work is intended to encourage analysis of student and teacher work to understand what alignment looks like. We have learned through this year s Common Core pilots that beginning with rigorous assessments drives significant shifts in curriculum and pedagogy. Universal Design for Learning (UDL) support is included to ensure multiple entry points for all learners, including students with disabilities and English language learners. PERFORMANCE TASK: ASSESSMENT 1.3 UNIVERSAL DESIGN FOR LEARNING (UDL) PRINCIPLES.10 RUBRIC 12 ANNOTATED STUDENT WORK 34 INSTRUCTIONAL SUPPORTS.57 UNIT OUTLINE..58 ARC OF LESSONS 1.64 ARC OF LESSONS 2.67 ARC OF LESSONS 3.69 LESSON GUIDES..71 SUPPORTS FOR ENGLISH LANGUAGE LEARNERS SUPPORTS FOR STUDENTS WITH DISABILITIES Acknowledgements: The unit outline was developed by David Wearley and Lorraine Pierre with input from Curriculum Designers Alignment Review Team, Onifa Hutchinson, and William Hillburn. The tasks were developed by the NYC DOE Middle School Performance Based Assessment Pilot Design Studio Writers, in collaboration with the Institute for Learning. 2

4 2011 NYC Grade 6 Assessment 1 Performance Based Assessment 1 Proportional Reasoning Grade 6 Student Name School Date Teacher 4

5 2011 NYC Grade 6 Assessment 1 1. Giovanni is visiting his grandmother who lives in an apartment building on the 25th floor. Giovanni enters the elevator in the lobby, which is the first floor in the building. The elevator stops on the 16th floor. What percentage of 25 floors does Giovanni have left to reach his grandmother's floor? Use pictures, tables or number sentences to solve the task. Explain your reasoning in words. 5

6 2011 NYC Grade 6 Assessment 1 2. Pianos and pipe organs contain keyboards, a portion of which is shown below. a) What is the ratio of black keys to white keys in the picture above? b) If the pattern shown continues, how many black keys appear on a portable keyboard with 35 white keys? c) If the pattern shown continues, how many black keys appear on a pipe organ with a total of 240 keys? 6

7 2011 NYC Grade 6 Assessment 1 3. a. Mr. Copper s class has a female student to male student ratio of 3:2. Mr. Copper s class has 18 girls, how many boys does he have? Show how you determined your answer. Explain your reasoning in words. b) Ms. Green s class has the same number of students as Mr. Copper s class. Her female-to-male ratio is 2:1. Which class has the greater number of females? How do you know? 7

8 2011 NYC Grade 6 Assessment 1 4. Use the recipe shown in the table to answer the questions below. Use pictures, tables or number sentences to solve the task. Grandma s Recipe for Sugar Cookies 1 ½ cups butter 2 cups sugar 4 eggs ¾ teaspoon baking powder 1 ¼ cups flour ¼ teaspoon salt a) How many cups of sugar are needed for each egg? How do you know? b) Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words. 8

9 2011 NYC Grade 6 Assessment 1 5. Fashion designers are trying to decide on just the right shade of blue for a new line of jeans. They have several bottles of fabric color, some with blue color and some with white color. They plan to mix these together to get the desired color. Mix A = { } Mix B = { } a) Will both mixes produce the same color jeans? Use mathematical reasoning to justify your answer. b) A designer uses the table below to think about her own special mix, Mix C. How many liters of blue color will she need to make a total of 40 liters? Explain your reasoning in words. Liters of Blue Color Liters of White Color Total Liters 9

10 GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS UNIVERSAL DESIGN FOR LEARNING (UDL) PRINCIPLES 10

12 GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS BENCHMARK PAPERS WITH RUBRICS This section contains benchmark papers that include student work samples for each of the tasks in the Ratios and Proportional Relationships assessment. Each paper has descriptions of the traits and reasoning for the given score point, including references to the Mathematical Practices. 12

13 NYC Grade 6 Assessment 1 Elevator Task Benchmark Papers 1. Giovanni is visiting his grandmother who lives in an apartment building on the 25th floor. Giovanni enters the elevator in the lobby, which is the first floor in the building. The elevator stops on the 16th floor. What percentage of 25 floors does Giovanni have left to reach his grandmother's floor? Use pictures, tables or number sentences to solve the task. Explain your reasoning in words. 13

14 NYC Grade 6 Assessment 1 Elevator Task Benchmark Papers 3 Points The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratio and percent. Minor arithmetic errors may be present, but no errors of reasoning appear. Complete explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with partwhole, ratio and percent). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a ratio and/or decimal and percent. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Understanding that the whole is 25 and either 16 or (25 16) floors is a part of the whole involved. Some students may consider a missing 13 th floor. In that case, the whole is 24 and either 15 or (24 15) is a part of the whole involved. b. Recognition of the need to determine how many out of 100. i. Using a fraction and its conversion to an equivalent fraction, possibly by simplifying first. ii. Converting to decimal then to percent. iii. Creating and reasoning from a grid representation of the context c. Recognition of the need either to work with 9/25 (or 9/24) or to subtract from 100% after changing 16/25 (or 15/24) to percent. 14

15 NYC Grade 6 Assessment 1 Elevator Task Benchmark Papers 2 Points The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratio and percent. Reasoning may contain incomplete, ambiguous or misrepresentative ideas. Partial explanations are stated based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with partwhole, ratio and percent). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a ratio and/or decimal and percent. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Understanding that the whole is 25 (or 24) and either 16 (or 15) floors is a part of the whole but failing to recognize the need to use 9/25 (or 9/24) or to subtract from 100% after changing 16/25 (or 15/24) to percent. b. Recognition of the need to work with 9/25 (or 9/24), but failing to change the ratio into a percent. 15

16 1 Point NYC Grade 6 Assessment 1 Elevator Task Benchmark Papers The response demonstrates some evidence of mathematical knowledge that is appropriate to the intent of the prompted purpose. An effort was made to accomplish the task, but with little success. Evidence in the response demonstrates that, with instruction, the student can revise the work to accomplish the task. Some evidence of reasoning is demonstrated either verbally or symbolically, but may be based on misleading assumptions, and/or contain errors in execution. Some work is used to find ratio and percent or partial answers are evident. Explanations are incorrect, incomplete or not based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with part-whole, ratio and percent). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a ratio and/or decimal and percent. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Forming the ratio 16/25 (or 15/24), and trying but failing to change the ratio into a percent. b. Forming the ratio 9/16 (or 9/24) and changing that ratio to a percent. 16

17 NYC Grade 6 Assessment 1 Piano Task Benchmark Papers 2. Pianos and pipe organs contain keyboards, a portion of which is shown below. a) What is the ratio of black keys to white keys in the picture above? b) If the pattern shown continues, how many black keys appear on a portable keyboard with 35 white keys? c) If the pattern shown continues, how many black keys appear on a pipe organ with a total of 240 keys? 17

18 NYC Grade 6 Assessment 1 Piano Task Benchmark Papers 3 Points The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve each part of the problem is stated, as is the work used to find ratios, proportions, and partial answers to problems. Minor arithmetic errors may be present, but no errors of reasoning appear. Accurate reasoning processes demonstrate the Mathematical Practice, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with tables and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation, proper symbolism and proper labeling of quantities. Repeated use of the same structure to solve parts b and c can be seen as evidence of the Mathematical Practice, (8) Look for and express regularity in repeated reasoning. The reasoning used to solve the parts of the problem may include: a. Scaling up the 5:7 ratio in fraction form to a denominator of 35, signifying 35 white keys. b. Scaling up the 5:7 ratio in tabular form. c. Using a proportion or proportional reasoning (e.g., 35 white keys is 5 times 7 white keys, so I can multiply 5 by 5 black keys to find the number of black keys). d. Recognizing the part-to-whole nature of part c as 5:12 and employing any of the above techniques

19 NYC Grade 6 Assessment 1 Piano Task Benchmark Papers 2 Points The response demonstrates adequate evidence of the learning and strategic tools necessary to complete the prompted purpose. It may contain overlooked issues, misleading assumptions, and/or errors in execution. Evidence in the response demonstrates that the student can revise the work to accomplish the task with the help of written feedback or dialogue. Either verbally or symbolically, the strategy used to solve each part of the problem is stated, as is the work used to find ratios, proportions, and partial answers to problems. Minor arithmetic errors may be present. Reasoning may contain incomplete, ambiguous or misrepresentative ideas. Accurate reasoning processes demonstrate the Mathematical Practice, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with tables and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation, proper symbolism and proper labeling of quantities. Repeated use of the same structure to solve parts b and c can be seen as evidence of the Mathematical Practice, (8) Look for and express regularity in repeated reasoning. The reasoning used to solve the parts of the problem may include: a. Scaling up the 5:7 ratio in fraction form to some denominator, but failing to stop at , signifying 35 white keys b. Scaling up the 5:7 ratio in tabular form, but failing to stop at 35. c. Reversing white and black keys, but maintaining the scaling to 35 white keys as indicated by labels. d. Using a proportion or proportional reasoning (e.g., 35 white keys is 5 times 7 white keys), but failing to then multiply 5 by 5 black keys to find the total number of black keys. e. Recognizing the part-to-whole nature of part c as 5:12 and employing any of the above techniques or errors. 19

20 1 Point NYC Grade 6 Assessment 1 Piano Task Benchmark Papers The response demonstrates some evidence of mathematical knowledge that is appropriate to the intent of the prompted purpose. An effort was made to accomplish the task, but with little success. Evidence in the response demonstrates that, with instruction, the student can revise the work to accomplish the task. Some evidence of reasoning is demonstrated either verbally or symbolically, is often based on misleading assumptions, and/or contains errors in execution. Some work is used to find ratios or proportions or partial answers to portions of the task are evident. Accurate reasoning processes demonstrate the Mathematical Practice, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with tables and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation, proper symbolism and proper labeling of quantities. Repeated use of the same structure to solve parts b and c can be seen as evidence of the Mathematical Practice, (8) Look for and express regularity in repeated reasoning. The reasoning used to solve the parts of the problem may include: a. Identification of the 5:7 ratio in some form (including tabular). b. Some attempt to scale, but failure to maintain the ratio, typically by reverting to addition. c. Failure to attempt at least two parts of the problem. 20

21 NYC Grade 6 Assessment 1 Boys and Girls Task Benchmark Papers 3. a) Mr. Copper s class has a female student-to-male student ratio of 3:2. If Mr. Copper s class has 18 girls, how many boys does he have? Show or explain in writing how you determined your answer. b) Ms. Green s class has the same number of students as Mr. Copper s class. Her female-to-male ratio is 2:1. Which class has the greater number of females? How do you know? 21

22 3 Points NYC Grade 6 Assessment 1 Boys and Girls Task Benchmark Papers The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratio and scaling. Minor arithmetic errors may be present, but no errors of reasoning appear. Complete explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practice (1) Make sense of problems and persevere in solving them and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with a table and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Scaling up the 3:2 ratio in fraction form to a numerator of 18, signifying 18 girls. Recognizing as the whole, and 2:1 as the ratio of girls-towhole in the other class, and following another scaling-up process b. Scaling up the 3:2 ratio in tabular form; scaling up the 2:1 ratio in a similar fashion, to a total of 30 students. c. Using a proportion or proportional reasoning (e.g., 18 girls is 6 times 3 girls, so I can multiply 6 by 2 boys to find the number of boys). Using a part-whole fraction to form a proportion for the second class. d. Recognizing that a 3:2 ratio is being compared with a 2:1 ratio for the same number of students. As a result, the ratio that represents a larger fraction also represents the class with the larger number of girls, since the ratio is female-to-male in both instances. a) Mr. Copper s class has a female student-to-male student ratio of 3:2. If Mr. Copper s class has 18 girls, how many boys does he have? Show or explain in writing how you determined your answer. b) Ms. Green s class has the same number of students as Mr. Copper s class. Her female-to-male ratio is 2:1. Which class has the greater number of females? How do you know? 22

23 2 Points NYC Grade 6 Assessment 1 Boys and Girls Task Benchmark Papers The response demonstrates adequate evidence of the learning and strategic tools necessary to complete the prompted purpose. It may contain overlooked issues, misleading assumptions, and/or errors in execution. Evidence in the response demonstrates that the student can revise the work to accomplish the task with the help of written feedback or dialogue. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratio and scaling. Reasoning may contain incomplete, ambiguous or misrepresentative ideas. Partial explanations are stated based on work shown. Accurate reasoning processes demonstrate the Mathematical Practice (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with a table and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Scaling up the 3:2 ratio in fraction or tabular form, but failing to stop at a numerator of 18, signifying 18 girls. Recognizing 2:1 as the ratio of girls to whole in the other class, and following another scaling up process (possibly incorrectly). b. Recognizing the need for a whole, but determining it with faulty reasoning or scaling part b to 18 girls rather than 30 students. c. Using a proportion or proportional reasoning (e.g., 18 girls is 6 times 3 girls, so I can multiply 6 by 2 boys to find the number of boys). Incorrectly using a part-part fraction to form a proportion for the second class. d. Recognizing that a 3:2 ratio is being compared with a 2:1 ratio for the same number of students; failing to determine the ratio that represents the larger fraction. a) Mr. Copper s class has a female student-to-male student ratio of 3:2. If Mr. Copper s class has 18 girls, how many boys does he have? Show how you determined your answer. Explain your reasoning in words. b) Ms. Green s class has the same number of students as Mr. Copper s class. Her female-to-male ratio is 2:1. Which class has the greater number of females? How do you know? 23

24 1 Point NYC Grade 6 Assessment 1 Boys and Girls Task Benchmark Papers The response demonstrates some evidence of mathematical knowledge that is appropriate to the intent of the prompted purpose. An effort was made to accomplish the task, but with little success. Evidence in the response demonstrates that, with instruction, the student can revise the work to accomplish the task. Some evidence of reasoning is demonstrated either verbally or symbolically, but may be based on misleading assumptions, and/or contain errors in execution. Some work is used to find ratio and scaling or partial answers are evident. Explanations are incorrect, incomplete or not based on work shown. The student may make some attempt at using some mathematical practices. Accurate reasoning processes demonstrate the Mathematical Practice (1) Make sense of problems and persevere in solving them and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with a table and/or ratios. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. Identification of the 3:2 ratio in some form (including tabular). b. Some attempt to scale, but failure to maintain the ratio, typically by reverting to addition. c. Failure to recognize that ratios are being compared in part b; attempting instead to compare by using the 3 from Mr. Copper s class with the 2 from Ms. Green s class (or 2 vs. 1). d. Failure to attempt at least one part of the problem a) Mr. Copper s class has a female student-to-male student ratio of 3:2. If Mr. Copper s class has 18 girls, how many boys does he have? Show how you determined your answer. Explain your reasoning in words. 24

25 NYC Grade 6 Assessment 1 Boys and Girls Task Benchmark Papers b) Ms. Green s class has the same number of students as Mr. Copper s class. Her female-to-male ratio is 2:1. Which class has the greater number of females? How do you know? 25

26 NYC Grade 6 Assessment 1 Sugar Cookies Task Benchmark Papers 4. Use the recipe shown in the table to answer the questions below. Use pictures, tables or number sentences to solve the task. Grandma s Recipe for Sugar Cookies 1 ½ cups butter 2 cups sugar 4 eggs ¾ teaspoon baking powder 1 ¼ cups flour ¼ teaspoon salt a) How many cups of sugar are needed for each egg? How do you know? b) Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words. 26

27 NYC Grade 6 Assessment 1 Sugar Cookies Task Benchmark Papers 3 Points The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve each part of the problem is stated, as is the work used to find ratios, unit rates, scaling and partial answers to problems. Minor arithmetic errors may be present, but no errors of reasoning appear. Complete explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with the tables and/or ratios). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a table, ratio or proportion. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. Evidence of the Mathematical Practice, (7) Look for and make use of structure, may be demonstrated by student recognition, without actually calculating, that, three times as much as can be interpreted as a 3/1 ratio. The reasoning used to solve the parts of the problem may include: a. A ratio of cups of sugar to each egg is formed and simplified or scaled down to a denominator of 1. b. Scaling down the 2:4 ratio in tabular form. c. Recognizing the phrase three times as much baking powder as salt as a 3:1 ratio. d. Forming a ¾ : ¼ ratio; possibly scaling that ratio up to 2 ¼ : ¾ on the basis of the three times as much language ½ 1 e. Drawing a picture and reasoning from the picture for either or both parts. ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ Baking Powder ¼ ¼ ¼ ¼ Salt 3 teaspoons baking powder : 1 teaspoon salt a. How many cups of sugar are needed for each egg? How do you know? b. Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words. 27

28 2 Points NYC Grade 6 Assessment 1 Sugar Cookies Task Benchmark Papers The response demonstrates adequate evidence of the learning and strategic tools necessary to complete the prompted purpose. It may contain overlooked issues, misleading assumptions, and/or errors in execution. Evidence in the response demonstrates that the student can revise the work to accomplish the task with the help of written feedback or dialogue. Either verbally or symbolically, the strategy used to solve each part of the problem is stated, as is the work used to find ratios, unit rates, scaling, and partial answers to problems. Minor arithmetic errors may be present. Reasoning may contain incomplete, ambiguous or misrepresentative ideas. Partial explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with the tables and/or ratios). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a table, ratio or proportion. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. Evidence of the Mathematical Practice, (7) Look for and make use of structure, may be demonstrated by student recognition, without actually calculating, that, three times as much as can be interpreted as a 3/1 ratio. The reasoning used to solve the parts of the problem may include: a. A ratio of cups of sugar to each egg is formed but not scaled down to a denominator of 1. b. A ratio of eggs to cups of sugar is formed and scaled down to a denominator of 1. c. Forming a ¾ : ¼ ratio; but incorrectly attempting to multiply by 3 on the basis of the three times as much language. d. Drawing a picture for both parts, but reasoning incorrectly from the picture for one part. a. How many cups of sugar are needed for each egg? How do you know? b. Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words. 28

29 1 Point NYC Grade 6 Assessment 1 Sugar Cookies Task Benchmark Papers The response demonstrates some evidence of mathematical knowledge that is appropriate to the intent of the prompted purpose. An effort was made to accomplish the task, but with little success. Evidence in the response demonstrates that, with instruction, the student can revise the work to accomplish the task. Some evidence of reasoning is demonstrated either verbally or symbolically, is often based on misleading assumptions, and/or contains errors in execution. Some work is used to find ratios or unit rates or partial answers to portions of the task are evident. Explanations are incorrect, incomplete or not based on work shown. Accurate reasoning processes demonstrate the Mathematical Practices, (1) Make sense of problems and persevere in solving them and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with the tables and/or ratios). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem as a table, ratio or proportion. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio notation and proper labeling of quantities. Evidence of the Mathematical Practice, (7) Look for and make use of structure, may be demonstrated by student recognition, without actually calculating, that, three times as much as can be interpreted as a 3/1 ratio. The reasoning used to solve the parts of the problem may include: a. Expressing a correct ratio, with no explanation. b. Forming the ratio of eggs to cups of sugar, but not scaling down to a denominator of 1. c. Forming a ratio from two ingredients, only one of which is baking powder or salt; however, correctly placing the baking powder or salt in the ratio. d. Drawing a picture, but reasoning incorrectly from the picture for both parts. a) How many cups of sugar are needed for each egg? How do you know? b) Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words. 29

30 NYC Grade 6 Assessment 1 Mixture Task Benchmark Papers 5. Fashion designers are trying to decide on just the right shade of blue for a new line of jeans. They have several bottles of fabric color, some with blue color and some with white color. They plan to mix these together to get the desired color. Mix A = { } Mix B = { } a) Will both mixes produce the same color jeans? Justify your reasoning. b) A designer uses the table below to think about her own special mix, Mix C. How many liters of blue color will she need to make a total of 40 liters? Explain your reasoning. Liters of Blue Color Liters of White Color Total Liters 30

31 3 Points NYC Grade 6 Assessment 1 Mixture Task Benchmark Papers The response accomplishes the prompted purpose and effectively communicates the student's mathematical understanding. The student's strategy and execution meet the content (including concepts, technique, representations, and connections), thinking processes and qualitative demands of the task. Minor omissions may exist, but do not detract from the correctness of the response. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratios, unit rates, scaling, and partial answers to problems. Minor arithmetic errors may be present, but no errors of reasoning appear. Complete explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practice, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem tables, ratios and/or proportions. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. Justification may include reasoning as follows: a. Forming unit rates and indicating that the unit rates differ or are not equivalent. b. Forming ratios and indicating that the ratios differ or are not equivalent. c. Building a table of values to a common number and noting, e.g., that one mix uses 9 blue and 6 white while the other differs, using 8 blue and 6 white, so the first is bluer d. Extending the table to a total of 40 and correctly identifying the number of blue liters. e. Using a proportion or proportional reasoning (e.g., 40 total liters is 5 times 8, so I can multiply 8 by 5 to find the number of blue liters). 31

32 2 Points NYC Grade 6 Assessment 1 Mixture Task Benchmark Papers The response demonstrates adequate evidence of the learning and strategic tools necessary to complete the prompted purpose. It may contain overlooked issues, misleading assumptions, and/or errors in execution. Evidence in the response demonstrates that the student can revise the work to accomplish the task with the help of written feedback or dialogue. Either verbally or symbolically, the strategy used to solve the problem is stated, as is the work used to find ratios, unit rates, scaling, and partial answers to problems. Reasoning may contain incomplete, ambiguous or misrepresentative ideas. Partial explanations are stated, based on work shown. Accurate reasoning processes demonstrate the Mathematical Practice (1) Make sense of problems and persevere in solving them and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem with tables, ratios and/or proportions. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. Justification may include reasoning as follows: a. Incorrectly forming or comparing unit rates or ratios for part a. b. Building a table of values to a common number, but incorrectly interpreting results for part a. c. Extending the table to 40 blue liters and/or a total of 40 liters, and possibly attempting to scale white liters to 40; therefore, incorrectly identifying the number of blue liters. d. Using a proportion or proportional reasoning incorrectly (e.g., 40 blue liters is 5 times 8, so I can multiply 8 by 5 to find the total number of liters). 32

33 1 Point NYC Grade 6 Assessment 1 Mixture Task Benchmark Papers The response demonstrates some evidence of mathematical knowledge that is appropriate to the intent of the prompted purpose. An effort was made to accomplish the task, but with little success. Evidence in the response demonstrates that, with instruction, the student can revise the work to accomplish the task. Some evidence of reasoning is demonstrated either verbally or symbolically, is often based on misleading assumptions, and/or contains errors in execution. Some work is used to find ratios, unit rates, scaling or partial answers to portions of the task are evident. Explanations are incorrect, incomplete or not based on work shown. The student may make some attempt at using some mathematical practices. Accurate reasoning processes demonstrate the Mathematical Practice, (1) Make sense of problems and persevere in solving them, and (2) Reason abstractly and quantitatively (since students need to abstract information from the problem, create a mathematical representation of the problem, and correctly work with both part-part and part-whole situations). Evidence of the Mathematical Practice, (3) Construct viable arguments and critique the reasoning of others, is demonstrated by complete and accurate explanations. Evidence of the Mathematical Practice, (4) Model with mathematics, is demonstrated by representing the problem tables, ratios and/or proportions. Evidence of the Mathematical Practice, (6) Attend to precision, can include proper use of ratio and/or proportion notation, proper symbolism and proper labeling of quantities. The reasoning used to solve the parts of the problem may include: a. No attempt at mathematical reasoning to respond to part a. b. Some attempt to scale in either or both parts, but failure to maintain the ratio, typically by reverting to addition. c. Incorrectly forming and comparing unit rates or ratios for part a; also, scaling blue, white and total in part b. 33

34 GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS ANNOTATED STUDENT WORK 34

35 NYC Grade 6 Assessment 1 Elevator Task Annotated Student Work Assessment 1: Question 1 1. Giovanni is visiting his grandmother who lives in an apartment building on the 24th floor. Giovanni enters the elevator in the lobby, which is the first floor in the building. The elevator stops on the 16th floor. What percentage of 24 floors does Giovanni have left to reach his grandmother's floor? Use pictures, tables or number sentences to solve the task. Explain your reasoning in words. CCSS (Content) Addressed by this Task: 6.RP.1 6.RP.3 6.RP.3c Understand the concept of ratio and use ratio language to describe a ratio relationship between two quantities. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent. CCSS for Mathematical Practice Addressed by this Task: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics (possible) 6. Attend to precision 35

36 NYC Grade 6 Assessment 1 Elevator Task Annotated Student Work Annotation of Student Work With a Score of 3 Content Standards: The student received a score of 3 because s/he recognizes that the context can be translated into a ratio of the number of floors traveled to number floors remaining (6.RP.1); however, the numbers are not explicitly labeled. The student then simplifies 16/24 to the equivalent ratio 2/3, which s/he appears to know to equal 66 2/3% (6.RP.3c). The student is able to explain the reasoning that s/he uses to find the solution to the problem. The student also correctly reasons that the percent remaining can be determined by subtracting the percent traveled from 100%. Mathematical Practices: This student makes sense of the problem (Practice 1) and is able to model the situation abstractly and quantitatively and relate the answer back to the context percentage of floors left (Practices 2 & 4). The student correctly expresses ratio equivalence (16/24 = 2/3), and uses the percent symbol when expressing the results (Practice 6). S/he presents a coherent explanation of the rationale for the procedure used (Practice 3). Next Instructional Steps: Ask the student to explain how s/he knows that 2/3 is equivalent to %. Ask the student how s/he can determine the percent of floors traveled if there were 25 floors instead of 24 floors. Ask him/her how this can be determined if the ratio is 15/23. 36

37 NYC Grade 6 Assessment 1 Elevator Task Annotated Student Work Annotation of Student Work With a Score of 2 Content Standards: The student received a score of 2 because s/he shows an understanding of ratio as a relationship between two quantities (6.RP.1) by creating a table with labeled columns. The student also shows an understanding that percent is a rate per 100 (6.RP.3c) when s/he uses a table to scale up 16/24 (6.RP.3) to find that it is equivalent to a rate of 64/96, which is approximated as 64%. The student also correctly reasons that the percent remaining can be determined by subtracting the percent traveled from 100%. However, the student chooses a strategy that, while it can be used in all situations, will generally not yield an exact answer. Mathematical Practices: This student makes sense of the problem (Practice 1) and is able to model the situation abstractly and quantitatively but does not explicitly relate the answer back to the context percentage of floors left (Practices 2 & 4). The student correctly expresses ratio equivalence, uses the percent symbol when expressing his results and correctly labels the columns in the table (Practice 6). While the symbolic work is well-organized and easy to follow, the student does not present a written explanation of the rationale for the procedure used (Practice 3). Next Instructional Steps: Ask the student to justify how s/he knows that Giovonni had traveled approximately 64% of the floors, and ask whether the exact percent would be more, or less, than 64%. Ask the student how s/he knows that 12 floors is equivalent 50%. Finally, ask the student to think of a way that s/he can find the exact percent of floors remaining. If the student is unable to make progress, suggest that the student join a group who used a different procedure, and have the students explain their strategy. 37

GRADE 8 MATH: TALK AND TEXT PLANS

GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculum-embedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems

GRADE 6 MATH: SHARE MY CANDY

GRADE 6 MATH: SHARE MY CANDY UNIT OVERVIEW The length of this unit is approximately 2-3 weeks. Students will develop an understanding of dividing fractions by fractions by building upon the conceptual

Understand the Concepts of Ratios and Unit Rates to Solve Real-World Problems

Grade 6 Mathematics, Quarter 2, Unit 2.1 Understand the Concepts of Ratios and Unit Rates to Solve Real-World Problems Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned

GRADE 6 MATH: GROCERY SHOPPING AND THE QUILT OF A MATH TEACHER

GRADE 6 MATH: GROCERY SHOPPING AND THE QUILT OF A MATH TEACHER UNIT OVERVIEW This unit contains a curriculum- embedded Common Core aligned task and instructional supports. The task is embedded in a 2 3

GRADE 5 MATH: STUFFED WITH PIZZA

GRADE 5 MATH: STUFFED WITH PIZZA UNIT OVERVIEW In this unit students will develop and expand the concept of rational numbers by using several interpretations and different types of physical models. TASK

SBI Lesson Objectives Common Core Standards MN State standards Lesson 1: Ratios

The Project MARS curriculum consists of 21 lessons designed to be implemented over a 6 week instructional period. The following chart outlines each lesson and how it aligns with Minnesota state standards

Grade Level/Course: Grades 6-7 Lesson/Unit Plan Name: Part 2 - Multiple Representations of Ratios: from concrete to operational. Rationale/Lesson Abstract: How comparing and contrasting multiple representations

TCAP/CRA Phase II Rectangle Task Anchor Set. SECURE MATERIAL - Reader Name: Tennessee Comprehensive Assessment Program

SECURE MATERIAL - Reader Name: Tennessee Comprehensive Assessment Program TCAP/CRA 2014 6 Phase II Rectangle Task Anchor Set Copyright 2014 by the University of Pittsburgh and published under contract

Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

New York State Testing Program Grade 4 Common Core Mathematics Test. Released Questions with Annotations

New York State Testing Program Grade 4 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY

#1 Make sense of problems and persevere in solving them.

#1 Make sense of problems and persevere in solving them. 1. Make sense of problems and persevere in solving them. Interpret and make meaning of the problem looking for starting points. Analyze what is

Common Core Standards for Mathematics Alignment to Early Numeracy

Common Core Standards for Mathematics Alignment to Early Numeracy The Common Core State Standards (http://www.corestandards.org/ ) provide a consistent, clear understanding of what students are expected

6th Grade Common Core State Standards. Flip Book

6th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices and the content standards and to get detailed information and

5 th Grade Common Core State Standards. Flip Book

5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at

This lesson introduces students to decimals.

NATIONAL MATH + SCIENCE INITIATIVE Elementary Math Introduction to Decimals LEVEL Grade Five OBJECTIVES Students will compare fractions to decimals. explore and build decimal models. MATERIALS AND RESOURCES

6th Grade Integrated ELD/Mathematics Three Phase Lesson

6th Grade Integrated ELD/Mathematics Three Phase Lesson PLANNING THE LESSON: DESIGNING INSTRUCTION FOR DISCIPLINARY THINKING AND LEARNING FOCUS QUESTION Foster Metacognition What is a ratio and how does

Fourth Grade Math Example SLO A Student Learning Objective (SLO) is a detailed process used to organize evidence of student growth over a specified period of time. The SLO process is appropriate for use

Composing and Decomposing Whole Numbers

Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Demonstrate understanding of mathematical

Total Student Count: 3170. Grade 8 2005 pg. 2

Grade 8 2005 pg. 1 Total Student Count: 3170 Grade 8 2005 pg. 2 8 th grade Task 1 Pen Pal Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Convert cake baking temperatures

Grade 4 Number & Operations in Base Ten 4.NBT.1-3

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 4 Number & Operations in Base Ten 4.NBT.1-3 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE

MINIATURE GOLF TARGET COMMON CORE STATE STANDARD(S) IN MATHEMATICS G.SRT.8

This task was developed by high school and postsecondary mathematics and design/pre-construction educators, and validated by content experts in the Common Core State Standards in mathematics and the National

Fourth Grade Mathematics Unit 2 Culminating Task: Pattern Block Puzzles STANDARDS FOR MATHEMATICAL CONTENT MCC4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by

Overview. Essential Questions. Grade 4 Mathematics, Quarter 3, Unit 3.3 Multiplying Fractions by a Whole Number

Multiplying Fractions by a Whole Number Overview Number of instruction days: 8 10 (1 day = 90 minutes) Content to Be Learned Understand that a fraction can be written as a multiple of unit fractions with

3 5-11) 3: & 1 #1 7.RP.1 1/2 1/4 1/2/1/4 2 3 #2 #3 #4 6 2 ½ 2 #5 (0, 0) (1, 4 #6 7.RP.3 7.G.1 #7 7.G.2 4: & 6 #1 7.RP.3 #2 7.RP.3, 7.SP.SP.1, 7.SP.

7 th Grade Math Quarter 3 (Jan 5- March 11) Unit 3: Ratios & Proportions Week 1 #1 7.RP.1 Calculate and interpret unit rates of various quantities involving ratios of fractions that contain like and different

New York State Testing Program Grade 3 Common Core Mathematics Test. Released Questions with Annotations

New York State Testing Program Grade 3 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY

Mathematics. Mathematical Practices

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with

Numbers and Operations

Grade 7 MATH Unit Number 1 Numbers and Operations Title Big Ideas/Enduring Understandings Problems can be solved easier when using variety of equivalent forms. All numbers fall within the real number system.

Performance Assessment Task Cindy s Cats Grade 5. Common Core State Standards Math - Content Standards

Performance Assessment Task Cindy s Cats Grade 5 This task challenges a student to use knowledge of fractions to solve one- and multi-step problems with fractions. A student must show understanding of

Overview. Essential Questions. Grade 5 Mathematics, Quarter 3, Unit 3.1 Adding and Subtracting Decimals

Adding and Subtracting Decimals Overview Number of instruction days: 12 14 (1 day = 90 minutes) Content to Be Learned Add and subtract decimals to the hundredths. Use concrete models, drawings, and strategies

Working with Equivalent Fractions, Decimals & Percentages

Virtual Math Girl: Module 3 (Student) and 4 (Teacher) Working with Equivalent s, Decimals & Percentages Activities and Supplemental Materials The purpose of Virtual Math Girl Module 3 (Student) and 4 (Teacher)

Mathematics. Curriculum Content for Elementary School Mathematics. Fulton County Schools Curriculum Guide for Elementary Schools

Mathematics Philosophy Mathematics permeates all sectors of life and occupies a well-established position in curriculum and instruction. Schools must assume responsibility for empowering students with

Conceptual Understanding of Operations of Integers

Grade 7 Mathematics, Quarter 2, Unit 2.1 Conceptual Understanding of Operations of Integers Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Demonstrate conceptual

HIGH SCHOOL ALGEBRA: THE CYCLE SHOP

HIGH SCHOOL ALGEBRA: THE CYCLE SHOP UNIT OVERVIEW This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The task is embedded in a 4-5 week unit on Reasoning with Equations

7 th Grade Unit 2: Ratios and Proportions (4 Weeks)

7 th Grade Unit 2: Ratios and Proportions (4 Weeks) Stage 1 Desired Results Established Goals Unit Description This unit focuses on analyzing proportional relationships and using them to solve real-world

Tennessee Department of Education Task: Chocolate Chips Task 4th Grade Lily and Anna are making chocolate chip cookies. They will need 4 cups of chocolate chips to make cookies for their friends. Lily

CORE Assessment Module Module Overview

CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve real-life and mathematical problems

Mathematics Grade-Level Instructional Materials Evaluation Tool

Mathematics Grade-Level Instructional Materials Evaluation Tool Quality Review GRADE 8 Textbooks and their digital counterparts are vital classroom tools but also a major expense, and it is worth taking

Comparing Fractions and Decimals

Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals Overview Number of Instructional Days: 10 (1 day = 45 60 minutes) Content to be Learned Explore and reason about how a number representing

Grade 7 Pre-AP Math Unit 1. Numbers and Operations *CISD Safety Net Standards: 7.3B. Suggested Time Frame. 1 st Six Weeks 15 Days

Numbers and Operations *CISD Safety Net Standards: 7.3B 7.3B Title Big Ideas/Enduring Understandings Problems can be solved easier when using variety of equivalent forms. All numbers fall within the real

Name Period Date MATHLINKS: GRADE 7 STUDENT PACKET 1 FRACTIONS AND DECIMALS

Name Period Date 7- STUDENT PACKET MATHLINKS: GRADE 7 STUDENT PACKET FRACTIONS AND DECIMALS. Terminating Decimals Convert between fractions and terminating decimals. Compute with simple fractions.. Repeating

Preview of Common Core State Standards Sample EAGLE Items Grade 7 Mathematics

Preview of Common Core State Standards Sample EAGLE Items Grade 7 Mathematics July 11, 2012 Grade 7 Technology enabled, multiple part, constructed response item types use a common context and contain several

DRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations

DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY,

GRADE 7 SKILL VOCABULARY MATHEMATICAL PRACTICES Add linear expressions with rational coefficients. 7.EE.1

Common Core Math Curriculum Grade 7 ESSENTIAL QUESTIONS DOMAINS AND CLUSTERS Expressions & Equations What are the 7.EE properties of Use properties of operations? operations to generate equivalent expressions.

COMMON CORE CROSSWALK

COMMON CORE CROSSWALK Grades K - 4 Mathematics Breakout Session January/February 2011 Missouri Department of Elementary and Secondary Education Agenda Common Focus CCSS Search What s common about the Mathematics

Division with Whole Numbers and Decimals

Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers

Just want the standards alone? You can find the standards alone at

5 th Grade Mathematics Unpacked Content This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these

Using Models to Understand Percents

Using Models to Understand Percents A Cent for Your Thoughts Lesson 20- Using Models to Understand Percents Learning Targets: Find a percent of a quantity as a rate per 00. Represent ratios and percents

Common Core State Standards. Standards for Mathematical Practices Progression through Grade Levels

Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for

8 as the number of objects in each share when 56 objects are partitioned equally. them. shares or a number of groups can be expressed as 56 8.

Tennessee Department of Education: Task: Matthew s Dilemma 3 rd Grade Matthew did not know the answer to 72 divided by 8. Are each of the following an appropriate way for Matthew to think about the problem?

PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

Common Core Standards Curriculum Map Mathematics - Grade 6

Common Core Standards Curriculum Map Mathematics - Grade 6 Quarter Unit Title Instructional Days 1 1 Factors, Multiples and Decimal Computation 12 2 Division of Fractions 13 3 Rates and Ratios 15 2 4/5

A REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Performance Assessment Task Hexagons Grade 7 task aligns in part to CCSSM HS Algebra The task challenges a student to demonstrate understanding of the concepts of relations and functions. A student must

Tennessee Department of Education Task: Birthday Party Task 3 rd Grade Ella s mother is baking 4 pans of brownies for a birthday party. Each pan can be divided into 16 squares of brownies. Ella wants to

Common Core State Standards Pacing Guide 1 st Edition. Math

Common Core State Standards Pacing Guide 1 st Edition Math Fourth Grade 3 rd Nine Week Period 1 st Edition Developed by: Christy Mitchell, Amy Moreman, Natalie Reno ``````````````````````````````````````````````````````````````````````````````````````

Fourth Grade. Operations & Algebraic Thinking. Use the four operations with whole numbers to solve problems. (4.OA.A)

Operations & Algebraic Thinking Use the four operations with whole numbers to solve problems. (4.OA.A) 4.OA.A.1: Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement

GRADE 1 MATH: NINA S NUMBERS

GRADE 1 MATH: NINA S NUMBERS UNIT OVERVIEW This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The task is embedded in a 2-3 week unit on Number and Operations in Base

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

Mathematics Common Core Cluster Mathematics Common Core Standard Resources. EE.6.1 Write and evaluate numerical expressions involving wholenumber

Unit 2 2 2 Unit : Operations and Properties Mathematics Common Core Domain Mathematics Common Core Cluster Mathematics Common Core Standard Resources NS..2 Fluently divide multi-digit numbers using the

GRADE 3 MATH: COOKIE DOUGH UNIT OVERVIEW This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The task is embedded in a 4-5 week unit on interpreting and linking representations,

A Correlation of. to the. Common Core State Standards for Mathematics Grade 4

A Correlation of to the Introduction envisionmath2.0 is a comprehensive K-6 mathematics curriculum that provides the focus, coherence, and rigor required by the CCSSM. envisionmath2.0 offers a balanced

Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

, Seventh Grade Math, Quarter 2

2016.17, Seventh Grade Math, Quarter 2 The following practice standards will be used throughout the quarter: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.

Grade Level Year Total Points Core Points % At Standard %

Performance Assessment Task Marble Game task aligns in part to CCSSM HS Statistics & Probability Task Description The task challenges a student to demonstrate an understanding of theoretical and empirical

HIGH SCHOOL GEOMETRY: COMPANY LOGO

HIGH SCHOOL GEOMETRY: COMPANY LOGO UNIT OVERVIEW This 3-4 week unit uses an investigation of rigid motions and geometric theorems to teach students how to verify congruence of plane figures and use the

Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. MP.

Performance Assessment Task Incredible Equations Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be able to understand

Grade 3 Measurement and Data 3.MD.5 & 6

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 3 Measurement and Data 3.MD.5 & 6 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATHTASKS Measurement

INFORMATIONAL GUIDE FOR THE MATHEMATICS PORTFOLIO APPEALS PROCESS

INFORMATIONAL GUIDE FOR THE MATHEMATICS PORTFOLIO APPEALS PROCESS The most important part of the PARCC Portfolio Appeal process is the evidence gathered to show the student s ability to demonstrate the

Upon successful completion of this course you should be able to

University of Massachusetts, Amherst Math 113 Math for Elementary School Teachers I Fall 2011 Syllabus of Objectives and Learning Outcomes Overview of the Course This is a mathematics content course which

PERCENTS. Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% =

PERCENTS Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% = Percents are really fractions (or ratios) with a denominator of 100. Any

Grade 3. Measurement and Data 3.MD.7.a-d 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 3 Measurement and Data 3.MD.7.a-d 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS Measurement

SAME BUT DIFFERENT?!?!?

SAME BUT DIFFERENT?!?!? MULTIPLE REPRESENTATIONS OF PROPORTIONALITY Sami Briceño Texas Lead Manager of School Partnerships INK YOUR THINKING What is Proportionality? How have you taught Proportionality

Number Sense and the Common Core

Number Sense and the Common Core Marilyn Burns MRI Math Reasoning Inventory Funded by the Bill & Melinda Gates Foundation Formative Assessment Find out what students really understand about math First

Mathematics Pacing Resource Document 5.AT.2 Standard: 5.AT.2: Teacher Background Information:

Standard: : Solve real-world problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models and equations

Math Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105

Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following

Problem of the Month: Perfect Pair

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

Ratios and Rates 6.RP.A Application Mini-Assessment by Student Achievement Partners

Ratios and Rates 6.RP.A Application Mini-Assessment by Student Achievement Partners OVERVIEW This mini-assessment is designed to illustrate some of the expectations of cluster 6.RP.A, which sets an expectation

Polygons and Area. Overview. Grade 6 Mathematics, Quarter 4, Unit 4.1. Number of instructional days: 12 (1 day = minutes)

Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area Overview Number of instructional days: 12 (1 day = 45 60 minutes) Content to be learned Calculate the area of polygons by composing into rectangles

Just want the standards alone? You can find the standards alone at

5 th Grade Mathematics Unpacked Content For the new Common Core State Standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North

Grade 3 3.OA.1. Operations and Algebraic Thinking 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS THE OFFICE OF MATHEMATICS Grade 3 Operations and Algebraic Thinking 3.OA.1 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

Vocabulary: Distributive property Simplify Like terms Solution Inverse operations

Targeted Content Standard(s): 8.EE.7 Solve linear equations in one variable. a) Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which

SGO Example: Mathematics, Grade 6 Overview This Student Growth Objective (SGO) was created by a 6th grade mathematics teacher to focus on Mathematical Practice 4 (MP4) Model with mathematics as articulated

7th Grade Mathematics Unit 1 Curriculum Map: September 9th October 25th

7th Grade Mathematics Unit 1 Curriculum Map: September 9th October 25th ORANGE PUBLIC SCHOOLS OFFICE OF CURRICULUM AND INSTRUCTION OFFICE OF MATHEMATICS Common Core Standards 6.NS.5 6.NS.6a 6.NS.7c REVIEW

Performance Assessment Task Peanuts and Ducks Grade 2. Common Core State Standards Math - Content Standards

Performance Assessment Task Peanuts and Ducks Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be fluent with addition

Whitnall School District Report Card Content Area Domains

Whitnall School District Report Card Content Area Domains In order to align curricula kindergarten through twelfth grade, Whitnall teachers have designed a system of reporting to reflect a seamless K 12

Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

5 th Grade Mathematics Instructional Week 20 Rectilinear area with fractional side lengths and real-world problems involving area and perimeter of 2-dimensional shapes Paced Standards: 5.M.2: Find the

TM parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN GRADE SIX 6 America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

Standards for Mathematical Practices Grade 6. Explanations and Examples

Standards for Mathematical Practices Grade 6 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments

Unit Plan Components. Appropriate Length of Time. Unit Name: Addition and Subtraction to 20. Common Core Standards

Unit Plan Components Big Goal Standards Big Ideas Unpacked Standards Scaffolded Learning Resources Appropriate Length of Time Aligned Assessment Comments This unit plan identifies a big goal for the overall

Classworks Common Core Transition Guide. South Carolina 4th Grade Mathematics

Classworks Common Core Transition Guide South Carolina 4th Mathematics OFFICIALLY ENDORSED Classworks Common Core Transition Guide - BETA Making The Transition With the increased rigor, the transition

Title: Building with Tiles CCGPS Standards Addressed: Grade: 8 Learning Goals: BIG Idea: Linear Patterns & Algebraic Thinking MCC8.F.2 Compare properties of two functions each represented in a different

Common Core State Standards

Common Core State Standards Common Core State Standards 7.NS.2 7.NS.2d 7.EE.3 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Tennessee Department of Education. Task: Jenna s Homework Problem 5 th Grade. Teacher Notes:

Tennessee Department of Education Task: Jenna s Homework Problem 5 th Grade Jenna completed 2/3 of the math problems on her homework during her stay at Fun Company after school. Afterward, she finished