4th Grade - Gateway 3

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• In Teacher Tools, Math Teacher Tools, Preparing to Teach Fishtank Math, Preparing to Teach a Math Unit recommends seven steps for teachers to prepare to teach each unit as well as the questions teachers should ask themselves while preparing. For example step 1 states, “Read and annotate the Unit Summary-- Ask yourself: What content and strategies will students learn? What knowledge from previous grade levels will students bring to this unit? How does this unit connect to future units and/or grade levels?”

• In Unit 2, Multi-Digit Multiplication, Unit Summary provides an overview of content and expectations for the unit. Within Unit Prep, Intellectual Prep, there is Unit-Specific Intellectual Prep detailing the content for teachers. The materials state, “Read pp. 14–15 in Progressions for the Common Core State Standards in Mathematics Number and Operations in Base Ten, K-5 (starting at the section titled ‘Use place value understanding and properties of operations to perform multi-digit arithmetic’). Read the document ‘Situation Types for Operations in Word Problems’ for multiplication and division. Identify the word problem types of any applicable assessment questions. Read the following table that includes models used throughout the unit.” Additionally, the Unit Summary contains Essential Understandings. It states, “In an additive comparison, the underlying question is what amount would be added to one quantity in order to result in the other. In a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other. One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1,000. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property. Rounding numbers can help one to determine whether an answer is reasonable based on whether the estimate is close to the computed answer or not. Making sense of problems and persevering to solve them is an important practice when solving word problems. Keywords do not always indicate the correct operation.”  

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Teacher Tools, Math Tools, Preparing to Teach Fishtank Math, Components of a Math Lesson states, “Each math lesson on Fishtank consists of seven key components: Objective, Standards, Criteria for Success, Tips for Teachers, Anchor Tasks/Problems, Problem Set, and Target Task. Several components focus specifically on the content of the lesson, such as the Standards, Anchor Tasks/Problems, and Target Task, while other components, like the Tips for Teachers, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” Examples include:

• In Unit 2, Multi-Digit Multiplication, Lesson 2, Anchor Tasks, Problem 1 Notes provide teachers guidance about how to set students up to solve the problems. The materials state, “Students may write a multiplication sentence (3 × ___= 24 or 24 ÷ 3 = ___). Ensure they see these as equivalent. Further, push them to use a letter to represent the unknown to connect back to work in Unit 1. An example tape diagram is shown below.” 

• In Unit 3, Multi-Digit, Division, Lesson 5, Tips for Teachers provide context about representations of problems using two-digit dividend division problems. The materials state, “Throughout this lesson, students may want to divide starting with the smallest place value. This strategy will work in this lesson, but students will see in Lesson 5 why starting with the largest place value first is the most effective way to divide. Therefore, model starting with the largest place value here, but don’t force students to do so. In Lesson 5, they will develop that understanding.”

• In Unit 6, Decimal Fractions, Lesson 4, Tips for Teachers include guidance for teachers to review key concepts about representing decimal place value. The materials state, “There are several ways to read decimals aloud. For example, 0.15 can be read as ‘1 tenth and 5 hundredths’ or ‘15 hundredths,’ just as 1,500 is sometimes read ‘15 hundred’ or ‘1 thousand, 5 hundred’ (NF Progression, p. 15). In your own language use, opt for the former way of saying a decimal, since this is more common and because reading each place value separately will become cumbersome (e.g., ‘4.583’ would be read as ‘four and 5 tenths 8 hundredths 3 thousandths’ rather than ‘four and 583 thousandths’). While ‘mathematicians and scientists often read 0.15 aloud as “zero point one five” or “point one five,”’ refrain from using this language until you are sure students have a strong sense of place value with decimals (NF Progression, p. 15).”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.  

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the course summary standards map, unit summary lesson map, and within each lesson. Examples include:

• In 4th Grade Math, Standards Map includes a table with each grade-level unit in columns and aligned grade-level standards in the rows. Teachers can easily identify a unit when each grade-level standard will be addressed. 

• In 4th Grade Math, Unit 2, Multi-Digit Multiplication, Lesson Map outlines lessons, aligned standards, and the objective for each lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• In Unit 7, Conversions, Lesson 7, the Core Standard is identified as 4.MD.A.1 and 4.MD.A.2. The Foundational Standard is identified as 3.MD.A.2 and 4.OA.A.2. Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Tasks, Problem Set & Homework, Target Task, and Additional Practice. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each Unit Summary includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Unit Summary includes an overview of how the math of this unit builds from previous work in math. The materials state, “Students understanding of the base ten system begins as early as Kindergarten, when students learn to decompose teen numbers as ten ones and some ones (K.NBT.1). This understanding continues to develop in Grade 1, when students learn that ten is a unit and therefore decompose teen numbers into one ten (as opposed to ten ones) and some ones and learn that the decade numbers can be referred to as some tens (1.NBT.2). Students also start to compare two-digit numbers (1.NBT.3) and add and subtract within 100 based on place value (1.NBT.4—6). In second grade, students generalize the place value system even further, understanding one hundred as a unit (2.NBT.1) and counting, reading, writing, comparing, adding, and subtracting numbers within 1,000 (2.NBT.2—9). In Grade 3, place value standards are additional cluster content, but they still spend time fluently adding and subtracting within 1,000 and rounding three-digit numbers to the nearest 10 and 100 (3.NBT.1—2).”

• In Unit 7, Unit Conversions, Unit Summary includes an overview of how the content in Grade 4 connects to mathematics students will learn in Grade 5 and the middle grades. The materials state, “As mentioned previously, the unit summarizes and applies much of the major work of the grade in the fresh context of measurement, serving as a nice culmination for the year. But, the unit not only looks back on the year in review, it also prepares students for work in future grades, including the very direct link to converting from smaller units to larger ones in Grade 5 (5.MD.1) and also to ratios and proportions in the middle grades (6.RP.1) as well as many other areas to come.”

• In Unit 8, Shapes and Angles, Unit Summary includes an overview of the Math Practices that are connected to the content in the unit. The materials state, “This unit allows for particular focus on MP.2, MP.5 and MP.6. For example, when students are “shown two sets of shapes and asked where a new shape belongs,” they are reasoning abstractly and quantitatively (MP.2) (G Progression, p. 16). Students also learn to use a new tool, the protractor, precisely, ensuring they line up the vertex and base correctly and read the angle measure carefully (MP.5, MP.6).”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. This information can be found within Our Approach and Math Teacher Tools. Examples where materials explain the instructional approaches of the program include:

• In Fishtank Mathematics, Our Approach, Guiding Principles include the mission of the program as well as a description of the core beliefs. The materials state, “Content-Rich Tasks, Practice and Feedback, Productive Struggle, Procedural Fluency Combined with Conceptual Understanding, and Communicating Mathematical Understanding.” Productive Struggle states, “We believe that students develop essential strategies for tackling complex problems, and build non-cognitive skills such as perseverance and resilience, through productive struggle. Productive struggle happens when students are asked to use multiple familiar concepts and procedures in unfamiliar applications, and the process for solving problems is not immediately apparent. Productive struggle can occur, and should occur, in multiple settings: whole class, peer-to-peer, and individual practice. Through instruction and high-quality tasks, students can develop a toolbox of strategies, such as annotating and drawing diagrams, to understand and attack complex problems. Through discussion, evaluation, and revision of problem-solving strategies and processes, students build interest, comfort, and confidence in mathematics.”

• In Math Teacher Tools, Preparing To Teach Fishtank Math, Understanding the Components of a Fishtank Math Lesson helps to outline the purpose for each lesson component. The materials state, “Each Fishtank math lesson consists of seven key components, such as the Objective, Standards, Criteria for Success, Tips for Teachers, Anchor Tasks/Problems, Problem Set, the Target Task, among others. Some of these connect directly to the content of the lesson, while others, such as Tips for Teachers, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” 

• In Math Teacher Tools, Academic Discourse, Overview outlines the role discourse plays within Fishtank Math. The materials state, “Academic discourse is a key component of our mathematics curriculum. Academic discourse refers to any discussion or dialogue about an academic subject matter. During effective academic discourse, students are engaging in high-quality, productive, and authentic conversations with each other (not just the teacher) in order to build or clarify understanding of a topic.” Additional documents are provided titled, “Preparing for Academic Discourse, Tiers of Academic Discourse, and Strategies to Support Academic Discourse.” These guides further explain what a teacher can do to help students learn and communicate mathematical understanding through academic discourse.

While there are many research-based strategies cited and described within the Math Teacher Tools, they are not consistently referenced for teachers within specific lessons. Examples where materials include and describe research-based strategies:

• In Math Teacher Tools, Procedural Skill and Fluency, Fluency Expectations by Grade states, “The language of the standards explicitly states where fluency is expected. The list below outlines these standards with the full standard language. In addition to the fluency standards, Model Content Frameworks, Mathematics Grades 3-11 from the Partnership for Assessment of Readiness for College and Careers (PARCC) identify other standards that represent culminating masteries where attaining a level of fluency is important. These standards are also included below where applicable. 4th Grade, 4.NBT.4, 4.NBT.2, 4.NBT.3, 4.NBT.5, 4.NBT.6, 4.NF.1, 4.NF.2, 4.NF.3c, 4.NF.4b, 4.NF.5, 4.NF.6, 4.NF.7, and 4.MD.1, among others.”

• In Math Teacher Tools, Academic Discourse, Tiers of Academic Discourse, Overview states, “These components are inspired by the book Classroom Discussions in Math: A Teacher’s Guide for Using Talk Moves to Support the Common Core and More. (Chapin, Suzanne H., Catherine O’Connor, and Nancy Canavan Anderson. Classroom Discussions in Math: A Teacher’s Guide for Using Talk Moves to Support the Common Core and More, 3rd edition. Math Solutions, 2013.)”

• In Math Teacher Tools, Supporting English Learners, Scaffolds for English Learners, Overview states, “Scaffold categories and scaffolds adapted from ‘Essential Actions: A Handbook for Implementing WIDA’s Framework for English Language Development Standards,’ by Margo Gottlieb. © 2013 Board of Regents of the University of Wisconsin System, on behalf of the WIDA Consortium, p. 50. https://wida.wisc.edu/sites/default/files/resource/Essential-Actions-Handbook.pdf”

• In Math Teacher Tools, Assessments, Overview, Works Cited lists, “Wiliam, Dylan. 2011. Embedded formative assessment.” and “Principles to Action: Ensuring Mathematical Success for All. (2013). National Council of Teachers of Mathematics, p. 98.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 4th Grade Course Summary, Course Material Overview, Course Material List 4th Grade Mathematics states, “The list below includes the materials used in the 4th grade Fishtank Math course. The quantities reflect the approximate amount of each material that is needed for one class. For more detailed information about the materials, such as any specifications around sizes or colors, etc., refer toeach specific unit.” The materials include information about supplies needed to support the instructional activities. Examples include:

• Base ten blocks are used in Units 1, 2, 3, and 6, maximum of 18 thousands, 40 hundreds, 50 tens, and 30 ones per individual, pair, or group of students. In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 2, students use base ten blocks to write and understand place value. The materials state, “You will need base ten blocks (as many thousands as you can find and about 15 each of hundreds, tens, and ones) for this task.”

• A ten-sided die, a spinner of digits 0-9, or digit cards are used in Unit 1, one per teacher.

• Six-sided dice are used in Unit 2, one or two per student.  

• Rulers (with eighth-inch measurements and ideally with a 0 mark not flush with edge) are used in Units 5 and 8, one per student. 

• Paper circles, such as parchment paper baking circles, are used in Unit 8, five per student. 

• Buttons or other small objects of various diameters are used Unit 5, twenty per pair or group of students.

• A balance scale is used in Unit 7, one per teacher. In Unit 7, Unit Conversions, Lesson 6, students express customary weight measurements in terms of a smaller unit. The materials state, “Teachers will need a balance scale, a one-pound weight, and 16 one-ounce weights for this task. (optional: these materials may be helpful for students to generate benchmarks that have that approximate weight.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for having assessment information included in the materials to indicate which standards and mathematical practices are assessed. 

Mid- and Post-Unit Assessments within the program consistently and accurately reference grade-level content standards and Standards for Mathematical Practice in Answer Keys or Assessment Analysis. Mid- and Post-Unit Assessment examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 7 is aligned to 4.NBT.1 and states, “The number 234 is multiplied by 10. Which statement is true about the digit 2 in the product? A. The value of the digit 2 in the product is 20. B. The value of the digit 2 in the product is 200. C. The value of the digit 2 in the product is 2,000. D. The value of the digit in the product is 20,000.”

• In Unit 3, Multi-Digit Division, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 1 is aligned to MP7 and states, “Fill in the blank below to make the equation true. 652\div4=(400\div4)+(__\div4)+(12\div4).”

• In Unit 5, Fraction Equivalence and Ordering, Unit Summary, Unit Assessment, Answer Key denotes standards addressed for each question. Problem 1 is aligned to 4.NF.1 and states, “Which fractions are equivalent to \frac{8}{12}? Select the three correct answers. A. \frac{2}{12}+\frac{4}{12}, B. \frac{6}{12} + \frac{2}{12}, C. \frac{1}{12} +\frac{2}{12} + \frac{4}{12} , D. \frac{3}{12}+\frac{2}{12}+\frac{2}{12}+\frac{1}{12} , E. \frac{1}{12} + \frac{2}{12} + \frac{1}{12} + \frac{2}{12} + \frac{1}{12} + \frac{2}{12}.” 

• In Unit 6, Decimal Fractions, Unit Summary, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 5b is aligned to MP3 and states, “Christy ran \frac{4}{10} mile on Monday and \frac{7}{100} mile on Tuesday. She said that she ran a total of \frac{47}{100} mile. Christy told Alex that she ran a greater distance than he ran, because 47 is more than 5. Identify the incorrect reasoning in Christy’s statement. Explain how Christy can correct her reasoning. Use >, <, or = to give a correct comparison between the distances that Alex and Christy ran.”

• In Unit 8, Unit Conversions, Unit Assessment Answer Key includes a constructed response and 2-point rubric with the aligned grade-level standard. Problem 4 is aligned to 4.MD.2 and states, “Mason ran for an hour and 15 minutes on Monday, 55 minutes on Tuesday, and 40 minutes on Wednesday. If Mason ran for 4 hours total Monday through Thursday, how long did he run for on Thursday? Show or explain your work.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each lesson provides a Target Task with a Mastery Response. According to the Math Teacher Tools, Assessment Overview states, “Target Tasks offer opportunities for teachers to gather information about what students know and don’t know while they are still engaged in the content of the unit.” Each Pre-Unit Assessment provides an answer key and guide with a potential course of action to support teacher response to data. Each Mid-Unit Assessment provides an answer key and a 2-, 3-, or 4-point rubric. Each Post-Unit Assessment Analysis provides an answer key, potential rationales for incorrect answers, and a commentary to support analysis of student thinking. According to Math Teacher Tools, Assessment Resource Collection,“commentaries on each problem include clarity around student expectations, things to look for in student work, and examples of related problems elsewhere on the post-unit assessment to look at simultaneously.” Examples from the assessment system include:

• In Unit 2, Multi-Digit Multiplication, Mid-Unit Assessment, Problem 5 states, “Compute 7,234 × 5. Explain how you know your answer is reasonable.” The answer key states, “36,170 and explanations may vary, e.g., ‘I know it’s reasonable because 7,234 ≈ 7,000 and 7,000 × 5 = 35,000, which is very close to my answer of 36,170.’” The 2-Point Scoring Rubric states, “2 points: Students response demonstrates an exemplary understanding of the concepts in the task. The student correctly and completely answers all aspects of the prompt. 1 point: Student response demonstrates a fair understanding of the concepts in the task. The student arrived at a partially acceptable conclusion, showing mixed evidence of understanding of the task, with some aspects of the task completed correctly, while others not. 0 points: Student response contains insufficient evidence of an understanding of the concepts in the task. Work may be incorrect, unrelated, illogical, or a correct solution obtained by chance.”

• In Unit 6, Decimal Fractions, Post Assessment Assessment Key, Question 5b, Correct Answer and Scoring Guidance states, “Valid explanation, e.g., ‘Christy found the correct total distance of her runs, but her comparison is wrong. 0.5 is \frac{5}{10} which equals \frac{50}{100} so she should compare 47 to 50, not 5. 50 is greater than 47, so \frac{5}{10} > \frac{47}{100} .’ OR ‘Christy's distance \frac{47}{100}=0.47 and Alex ran 0.5 mile, so she should compare 0.5 to 0.47. The 5 in tenths place in 0.5 has a greater value than the 4 in the tenths place in 0.47.’” The 2-point Scoring Rubric states, “2 points - Student response demonstrates an exemplary understanding of the concepts in the task. The student correctly and completely answers all aspects of the prompt. 1 point - Student response demonstrates a fair understanding of the concepts in the task. The student arrived at a partially acceptable conclusion, showing mixed evidence of understanding of the task, with some aspects of the task completed correctly, while others not. 0 points - Student response contains insufficient evidence of an understanding of the concepts in the task. Work may be incorrect, unrelated, illogical, or a correct solution obtained by chance.”

• In Unit 7, Lesson 8, Target Task, Problem 2 states, “Jacob needs to do his chores. It takes Jacob an hour and 45 minutes to mow the lawn and 20 minutes to clean his room. If he starts his chores at 2:00 pm, what time will he finish?” The Mastery Response includes, “1 hr 45 min = 105 min. 105 min. + 20 min. = 125 min.” A number line with marks at 2pm, 3pm, and 4pm is drawn depicting the solution at 4:05 pm.  

• In Unit 8, Shapes and Angles, Pre-Unit Assessment, Problem 3 states, “Draw a rectangle. Explain how you know the shape you drew is a rectangle. Understanding categories and attributes of shapes (3.G.A.1). In #2, students are asked to identify a quadrilateral that is not a trapezoid. This requires that students “have built a firm foundation of several shape categories, [which] can be the raw material for thinking about the relationships between classes. For example, students can form larger, superordinate, categories, such as the class of all the shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as squares, rectangles, rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of those subcategories (G Progressions, P.13). In #3, students are asked to draw a rectangle and describe how they know it is a rectangle. Students will rely on the understanding demonstrated in these two problems to classify shapes by properties of their lines as it relates to angle measure, they will classify shapes based on their presence or absences of these types of lines. Potential Course of Action - If needed, this concept should be reviewed before students analyze and classify shapes starting in Lesson 14. For example, include tasks similar to #2 above as a warmup to Lesson 14 and/or 16. If students struggled with #3, include tasks similar to it as a warmup to Lesson 15 and/ or 17. Find problems and other resources in the Fishtank lessons: Grade 3 Unit 5 Lesson 11-15.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The Expanded Assessment Package includes the Pre-Unit, Mid-Unit, and Post-Unit Assessments. While content standards are consistently identified for teachers within Answer Keys for each assessment, practice standards are not identified for teachers or students. Pre-Unit items may be aligned to standards from previous grades. Mid-Unit and Post-Unit Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, short answer, and constructed response. Examples include:

• In Unit 2, Multi-Digit Multiplication, Mid-Unit Assessment, Problems 1 and 4, and Post-Unit Assessment, Problems 3 and 5 develop the full intent of standard 4.OA.1 (Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations). Problem 1 states, “Which of the following statements can be represented by the equation 19×8=152? Select the two correct statements. A. 152 is 19 times as much as 8; B. 152 is 8 more than 19; C. 152 multiplied by 8 is the same as 19; D. 152 is 19 more than 8; E. 152 equals 8 times as many as 19; F. 152 added to 8 is 19.”  Problem 4 states, “A skyscraper is 27 stories tall. A house is 3 stories tall. Which equation could be used to find how many times higher the skyscraper is than the house? A. 3 × 27 = ___; B. 3 × ___ = 27; C. 3 + 27 = ___; D. 3 + ___ = 27.” Post-Unit Assessment, Problem 5 states, “Nathan sold n tickets for a school play. Joe also sold tickets for the school play. The number of tickets Joe sold, j, is shown by the equation below. 2×n = j. Which statement is true about selling tickets to the school play? A. Joe sold two more tickets than Nathan sold; B. Joe sold two fewer tickets than Nathan sold; C. Joe sold half as many tickets as Nathan sold; D. Joe sold twice as many tickets as Nathan sold.” Problem 3 states, “Steph has 40 rubber bands. She has 5 times as many rubber bands as Jake has. Which equation shows how to find the number of rubber bands Jake has? A. 40 + 5 = 45; B. 40 – 5 = 35; C. 40 × 5 = 200; D. 40 ÷ 5 = 8.” 

• In Unit 5, Fraction Operations, Post-Unit Assessment, Problem 8, supports the full development of MP2 (Reason abstractly and quantitatively, as students solve problems involving addition, subtraction, and multiplication of fractions). The materials state, “Omari wants to make fruit smoothies. The directions to make one smoothie include mixing \frac{4}{8} cup of yogurt and 1 cup of ice with the amounts of each fruit shown. \frac{5}{8} cup of banana slices, \frac{2}{8} cup of blueberries. Part A - Omari wants to make 6 smoothies. How many total cups of blueberries and banana slices will she use to make the 6 drinks? Show your work or explain your answer. Part B- Next Omari will add the yogurt and ice. How many total cups of yogurt and ice will she need to make the 6 smoothies? Show your work or explain your answer.”

• In Unit 6, Decimal Fractions, Post-Unit Assessment, Problems 5 and 8 support the full development of MP3 (Construct viable arguments and critique the reasoning of others, as students work with decimal fractions). Problem 5 states, “A. Alex ran 0.5 mile. What number should replace the ? to make a fraction equivalent to 0.5? \frac{?}{10}Write your answer below. B. Christy ran \frac{4}{10}mile on Monday and \frac{7}{100} mile on Tuesday. She said that she ran a total of \frac{47}{100} mile. Christy told Alex that she ran a greater distance than he ran, because 47 is more than 5. Identify the incorrect reasoning in Christy’s statement. Explain how Christy can correct her reasoning. Use >, <, or = to give a correct comparison between the distances that Alex and Christy ran.” Problem 8 states, “Jessica shades two grids that each equal one whole to represent and compare the fractions \frac{3}{10} and \frac{29}{100}. Part A. From the list below, select the decimal that represents \frac{3}{10} and the decimal that represents \frac{29}{100}and use them to create a true comparison. Answer Choices: 0.03; 0.3; 3.1; 0.29; 0.92; 2.9.  ___ > ___ Part B. Jessica says that \frac{3}{10}+ \frac{29}{100} = $$\frac{32}{100}$$ because 3 + 29 = 32 and there are 100 squares in each of the grids. Explain how you know Jessica is incorrect by using the grids or the decimal inequality you created. Then find the correct sum.”

• In Unit 7, Unit Conversions, Mid-Unit Assessment Problems 2 and 4, and Post-Unit Assessment Problem 4 develop the full intent of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit). Mid-Unit Assessment, Problem 2 states, “Kristy buys 3 pounds of chicken at the grocery store. She wants to split the chicken into four servings. How much chicken, in ounces, will each of Kristy’s servings be?” Problem 4 states, “Fernando has two bunches of string that are each 18 meters long. He uses 25 meters and 70 cm to make some bracelets. How much string, in millimeters, does Fernando have left?” Post- Unit Assessment, Problem 2 states, “A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How many milliliters of juice are left in the pitcher after filling the glass? A. 20 mL; B. 90 mL; C. 178 mL; D. 1,820 mL.” Problem 4 states, “Mason ran for an hour and 15 minutes on Monday, 55 minutes on Tuesday, and 40 minutes on Wednesday. If Mason ran for 4 hours total Monday through Thursday, how long did he run for on Thursday? Show or explain your work.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

There are no instances within the materials when advanced students have more assignments than their classmates, and there are opportunities where students can investigate grade-level mathematics at a higher level of complexity. Often, “Challenge” is written within a Problem Set or Anchor Task Guidance/Notes to identify these extensions. Examples include:

• In Unit 2, Multi-Digit Multiplication, Lesson 3, Homework, Problem 7 states, “CHALLENGE: Brown Squirrels can carry 2 acorns at a time. Gray Squirrels can carry 3 acorns at a time. Black Squirrels can carry 5 acorns at a time. Suppose the three squirrels all wanted to store acorns for the winter. Depending on how motivated each squirrel was, they would end up with different amounts. For instance suppose the Brown Squirrel took 4 trips, the Gray Squirrel took 2 trips and the Black Squirrel took 2 trips. The Brown Squirrel would end up with 8 acorns, the Gray Squirrel would have 6 acorns and the Black Squirrel would have 10. Between them they took every one of the 24 acorns. a. How many different ways could the three Squirrels divide up the 24 acorns and not leave any left over? Each Squirrel must carry his maximum load on each trip b. How do you know that you have found all of the ways?”

• In Unit 5, Fraction Operations, Lesson 11, Problem Set Problem 6b states, “CHALLENGE: Use Melissa’s strategy to compare \frac{24}{7} and \frac{31}{9}. Explain which fraction you chose for comparison and why.”

• In Unit 6, Decimal Fractions, Lesson 11, Tips For Teachers state, “This lesson is optional since it is not an explicit part of the standards for students to add decimals together. However, because this is simply a combination of skills explicitly outlined in the standards (namely, converting between fraction and decimal form (4.NF.6) as well as adding decimal fractions together (4.NF.5)) and because it will aid students’ work with solving problems involving money (4.MD.2), it is included here. It is at your discretion to keep or skip.”

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials provide suggestions and/or links for virtual and physical manipulatives that support the understanding of grade-level concepts. Manipulatives are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Examples include: 

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 5, Anchor Tasks, Problem 1 uses a Millions Place Value Chart as students multiply and divide single units by ten. The materials state, “For this task, students will need base ten blocks (10 ones, 10 tens, 10 hundreds, 1 thousand per group/teacher) and a Millions Place Value Chart.”

• In Unit 6, Decimal Fractions, Lesson 11, Tips for Teachers uses a game to reinforce hundredths written as decimals. The materials state, “As a supplement to the Problem Set, students can play the game Make Two by the San Francisco Unified School District Mathematics Department.”

• In Unit 8, Shapes and Angles, Lesson 2, Anchor Tasks, Problem 1 uses a shapes template to sort shapes into groups. The materials state, “Students will need Template: Shapes and paper (preferably circular paper, though not necessarily) for this task.”