6th Grade - Gateway 3

The materials reviewed for Fishtank Plus Math Grade 6 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 1, Understanding and Representing Ratios, Unit Summary provides an overview of content and expectations for the unit. Within Unit Prep, View Unit Launch, there is a video detailing the content for teachers. The materials state,  “Welcome to the Unit Launch for 6th Grade Math, Unit 1 Understanding and Representing Ratios. Please watch the video below to get started.” Additionally, the Unit Summary contains Intellectual Prep and Unit Launch with Standards Review, Big ideas, and Content connections. The Standards Review provides teachers an “opportunity to reflect on select standards from the unit. [...] In this section you will examine the language of these standards and reflect on how several problems on the end-of-unit assessment relate to the standard.”  Then, Big Ideas help teachers “understand how these ideas develop throughout the unit by analyzing lessons and problems from the unit, and finally, have the chance to reflect on how you will address your students’ needs around these concepts.” Finally, Content Connections states, “In this final section of the unit launch, you’ll have the chance to zoom out and look at the related content that students study before and after this unit.” This information is included for Units 1-5 in Grade 6.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. In 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 1, Understanding and Representing Ratios, Lesson 10, Tips for Teachers provide context about representations for ratios. The materials state, “Lessons 10 and 11 introduce students to using tables to represent equivalent ratios and solve problems. Students began this unit using discrete diagrams and pictures as representations. They then progressed to using double number lines, which allowed them to model with greater values. While tables provide greater flexibility than diagrams and double number lines, they are more abstract and can be difficult to interpret. If students struggle with understanding the relationships seen in tables, spiral back to double number lines and diagrams and make the connection between how ratios are seen in all three representations.”

• In Unit 2, Unit Rates and Percent, Lesson 7, Anchor Problem 2 Notes provide teachers guidance about how to set students up to solve the problem. The materials state, “Students need a conversion rate between miles and kilometers. This can be provided on a reference sheet given to students, or one or both of the rates can be given to the students as they work out the problem.”

• In Unit 4, Rational Numbers, Lesson 11, Tips for Teachers include guidance to address common misconceptions as students work with ordered pairs in a coordinate plane. The materials state, “A common misconception students may have prior to extending beyond the first quadrant is that an x−coordinate indicates how far to move right, and a y−coordinate indicates how far to move up. Discuss how the addition of negative values in an ordered pair creates the need for a more precise way to describe movement in the coordinate plane.”

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

Unit Summaries include a Unit Launch that provides a narrative overview of concepts within the unit and beyond, alongside sample problems and standard connections. Tips for Teachers, within some lessons, can also support teachers in developing a deeper understanding of course concepts. Examples include:

• In Unit 2, Unit Rates and Percent, Unit Summary, Intellectual Prep Unit Launch, Connections, Future Connections, contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. “Seventh grade students learn to compute complex fractions, where the numerator and/or the denominator of a fraction is also a fraction, a skill that they can use when finding the unit rate in the situation below. Lastly, students have the opportunity to apply their knowledge and skills around percent in 7th grade Unit 5, studying 7.RP.A.3. They go beyond solving for the part, whole, or percent, and consider real-world situations involving tax, fees, and simple interest. They also analyze how situations change by computing percent increases and percent decreases.”

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 4, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “Making Sense of Division of Fractions'' from SERP includes three great videos that demonstrate dividing with fractions using models and methods alternative to the general algorithm. They may be useful for your own reference or to use with students in smaller group settings. Additionally, Delaying "Invert and Multiply", also from SERP, is a valuable resource for teachers to deepen their understanding of fraction division and why the invert and multiply rule works prior to teaching the method to students.”

• In Unit 5, Numerical and Algebraic Expressions, Unit Summary, Intellectual Prep Unit Launch, Connections, Future Connections, contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. “In 7th grade, students pick up right where they left off in 6th grade with expressions, studying standards 7.EE.A.1, 7.EE.A.2, and 7.EE.B.3 in the Expressions and Equations domain. Students continue to focus on applying the properties of operations when working with expressions to either evaluate or generate equivalent expressions. As the expressions become more complex and multi-step, as well as involving rational numbers, students lean on their foundational work from prior grade-levels, as well as look for and make use of the structure of the expressions (Standard for Mathematical Practice 7). The two examples below from 7th grade, show how the problems become more advanced from 6th grade, and highlight how additional tools, such as the properties of operations and practice standards, can support students in their work. The first problem from Lesson 4 involves using the distributive property to expand and factor expressions with rational numbers. The second problem from Lesson 8 involves both expanding part of the expression as well as combining like terms to identify multiple equivalent expressions.”

• In Unit 8, Statistics, Lesson 1, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “This TEDx Talk video, "Why statistics are fascinating: the numbers are us" by Alan Smith may provide teachers will helpful perspective of why statistics is important to understand our populations and communities. (Note, this speaker is from the United Kingdom.)”

The materials reviewed for Fishtank Plus Math Grade 6 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 6th 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 6th Grade Math, Unit 2, Unit Rates and Percent, Lesson Map outlines lessons, aligned standards, and the lesson objective for each unit lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• In Unit 7, Geometry, Lesson 2, the Core Standard is identified as 6.G.A.1. The Foundational Standard is identified 4.MD.A.3. Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Problems, Problem Set, and Target Task. 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. Additionally, for Units 1 through 5, there is a Unit Launch within the Intellectual Prep that includes more information about content connections. Examples include:

• In Unit 2, Unit Rates and Percents, Unit Summary includes an overview of how the content in 6th grade builds from previous grades. The materials state, “In fourth and fifth grade, students interpreted fractions as division problems and began to make the connection between fractions and decimals. They multiplied fractions by whole numbers and other fractions in context of real-world problems, and they reasoned about what happens to a quantity when you multiply it by a number greater than one or less than one. Sixth grade students will draw on these prior skills and understandings as they make connections between unit rates and fractions, and between fractions, decimals, and percentages.” 

• In Unit 4, Rational Numbers, Unit Launch, Content Connections states, “While the concept of negative rational numbers is new to students, they have been working with the number line since elementary grades. In 2nd grade, students represent whole numbers on a number line (2.MD.B.6), connecting the idea of a point on a number line to the measurement of its length, as in a ruler. In 3rd grade, students develop an understanding of fractions as numbers (3.NF.A.2), and like whole numbers, represent fractions on a number line. This too is supported by their understanding of measurement. In addition to experience plotting positive numbers on the number line, students also have prior experience with plotting ordered pairs in the first quadrant of the coordinate plane (5.G.A.1 and 5.G.A.2). In 7th grade, students’ work with rational numbers naturally progresses to involve computations (7.NS.A). With a conceptual understanding of negative numbers, students are ready to consider what happens when you add, subtract, multiply, and divide these numbers. The work students do in this unit with the coordinate plane also sets them up for success working with linear relationships. In 6th and 7th grades, students represent proportional relationships in the first quadrant. In 8th grade, students begin to graph relationships in all four quadrants of the coordinate plane. In addition, 8th graders also develop an understanding of congruence and similarity of figures as they are transformed in the coordinate plane.”

• In Unit 5, Numerical and Algebraic Expressions, Unit Launch, Content Connections include prior and future standards with a narrative description of the connections. The materials state, “There are several standards in elementary grade-levels that build the early concept of variables and algebraic expressions. As early as Kindergarten students use numbers and operations to record the idea of composing and decomposing numbers, like 5 = 2 + 3 (K.OA.A.3), writing some of their first mathematical sentences. As early as 3rd grade, students use letters to represent unknown quantities in equations to solve problems (3.OA.D.8), and in 4th grade, students use letters and other symbols to indicate multiplicative comparison. For example, in the Target Task below, students select the equation that correctly represents the comparison between Sanjay and Luke’s granola bars, using the letter g to represent the number of granola bars Luke bought.”

• In Unit 6, Equations and Inequalities, Unit Summary does not contain a Unit Launch with Content Connections.

The materials reviewed for Fishtank Plus Math Grade 6 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 materials state, “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. 6th Grade, 6.NS.2, 6.NS.3, 6.NS.1.”

• 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 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 6th Grade Course Summary, Course Material Overview, Course Material List 6th Grade Mathematics states, “The list below includes the materials used in the 6th 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 to each specific unit.” The materials include information about supplies needed to support the instructional activities. Examples include:

• Four-Function calculators are used in Units 1, 2, 3, 7, and 8, one per student. 

• Graph paper is used in Units 3, 4, 6, and 7, one ream. 

• Markers are used in Unit 3, four to six sets, one set per small group.

• Patty paper is used in Unit 4, two or three sheets per student. In Unit 4, Rational Numbers, Lesson 4, students define opposites and label opposites on a number line. (Recognize that zero is its own opposite. 6.NS.C.6.A and 6.NS.C.6.B). Tips for Teachers states, “The following materials may be useful for this lesson: patty paper (transparency paper).” The notes for Anchor Problem 1 state, “If students are struggling, you can give them patty paper (transparency paper) to copy the number line and physically fold it along the 0 to see how the opposites line up. This may be especially helpful when working with non-integer rational numbers (MP.5). “

• A balance or mobile is used in Unit 6, one teacher set. 

• Play money is used in Unit 8, four to six sets, one set per small group. In Unit 8, Statistics, Lesson 5, Tips for Teachers states, “The following materials are helpful for this lesson: play money.”

• Unit 7 uses one teacher set of three-dimensional solids.

The materials reviewed for Fishtank Plus Math Grade 6 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, Understanding and Representing Ratios, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 2 is aligned to 6.RP.1 and states, “All of the benches in a park are red or blue. The ratio of red benches to blue benches in the park is 3:4. Based on this information, which of the following statements is true? a. For every 4 benches in the park, 3 are red. b. For every 7 benches in the park, 4 are red. c. For every 3 red benches in the park, there are 4 blue benches. d. For every 3 red benches in the park, there are 7 blue benches.”

• In Unit 2, Unit Rates and Percent, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 8 is aligned to MP3 and states, “At the grocery store, you can buy 2 avocados for $3, or you can buy a bag of 8 avocados for $10. Which buying option has the better unit price per avocado? Show or explain your reasoning.”

• In Unit 5, Numerical and Algebraic Expressions, Unit Summary, Unit Assessment, Answer Key denotes standards addressed for each question. Question 7 is aligned to 6.EE.3 and states, “Which quantity could go in the blank to make the equation true? 𝑥 + 2𝑥 ___ = 5𝑥. a. 2, b. 3, c. 2𝑥, d. 3𝑥.” 

• In Unit 6, Equations and Inequalities, Unit Summary, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 8 is aligned to MP4 and states, “The Jackson family is driving from Boston to New York for a summer vacation trip. So far, they have driven 84 miles which is 40% of the total miles of their trip. How many miles is the Jackson’s trip from Boston to New York? Write and solve an equation.”

• In Unit 7, Geometry, Unit Assessment Answer Key includes a constructed response and 2-point rubric with the aligned grade-level standard. Question 10 is aligned to 6.G.2 and states, “A farmer stacked hay bales. The length and width of each hay bale are shown below. The volume of each hay bale is 10\frac{2}{3}cubic feet. What is the height, in feet, of one hay bale?” An image of a bale of hay is shown with dimensions 4ft and 1\frac{1}{3} feet.

The materials reviewed for Fishtank Plus Math Grade 6 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 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.” Each Mid-Unit Assessment provides an answer key and a 1-, 2-, 3-, or 4-point rubric. Each Pre-Unit Assessment provides an answer key and guide with a potential course of action to support teacher response to data. 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.” Examples from the assessment system include:

• In Unit 2, Unit Rates and Percent, Post-Unit Assessment Analysis, Problem 10 states, “A notebook has a width of 8 inches. What is the width of the notebook in centimeters? Commentary: In this problem, students use ratio reasoning to convert inches to centimeters. Students may need to refer to a reference sheet to identify the correct unit rate to use between inches and centimeters. It may be valuable to refer to related multiple choice question #7 where students convert between yards and feet. Note, students are allowed to use calculators on this assessment, so the choice of operation and set up of the problem are more important than the actual calculation.”

• In Unit 4, Rational Numbers, Lesson 1, Target Task, students learn to extend the number line to include negative numbers. The materials state, “Problem 1, On the number line below, b is a negative integer and d is a positive integer. (on the number line, a is 2 units to the left of b. b is 4 units left of c. c is 3 units left of d) Name one set of values for a, b, c, and d. Problem 2, Explain the error in the number line below.” The Mastery Responses state, “Problem 1, answers vary. One possible answer: a = -3, b = -1, c = 3, and d = 6. Problem 2, The number line is numbered backwards. The numbers should get smaller as you go left and bigger as you go right.” 

• In Unit 6, Equations and Inequalities, Mid-Unit Assessment, Problem 5 states, “Teddy makes a stack of three blocks, placing one block on top of another. The first block is $$3\frac{1}{2}$$  inches tall. The second block is $$4\frac{3}{4}$$ inches tall. Teddy places the third block on the top of the other two, and the stack of blocks is now $$10\frac{1}{2}$$ inches tall. How tall is the third block? Write and solve an equation.” Tools for scoring purposes states, “The 2-point rubric below is used to score this problem. 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 8, Statistics, Pre-unit Assessment provides follow-up steps and suggestions that can guide the response to all students including students with special needs. There is a link to previously taught standards which states, “Decimal division and unit rates (6.NS.B.3, 6.RP.A.3).” There is task-specific guidance to help determine if a student has met the expectations. Problem 1 states, “In this unit, students apply their division skills to find the mean. Often, data sets will include decimal values testing students’ mastery of decimal operations. Students may notice that taking an average is computationally very much like finding a unit rate: students find (or are given) a total for a group of items and divide it by the number of items in the group. This is especially clear for students who master and internalize the idea of finding the mean as redistributing value among items in a dataset. Take note of the strategy students choose to divide; students should fluently use the standard algorithm at this point in the year.”

The materials reviewed for Fishtank Plus Math Grade 6 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, Unit Rates and Percent, Post-Unit Assessment, Problems 4 and 5 develop the full intent of standard 6.RP.2 (Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship). Problem 4 states, “A freight train traveled 144 miles in 6 hours. At what rate, in miles per hour, did the train travel?” Problem 5 states, “A group of students organized a car wash to raise money for a local charity. The students charged $5.00 for each car they washed. In 3 hours, they washed 12 cars. a. How many cars per hour did the students wash? b. How many hours per car did it take the students to wash? c. If the students continue at the same rate, how much money could they earn from washing cars for 8 hours?”

• In Unit 4, Rational Numbers, Mid-Unit Assessment Problem 6 and Post-Unit Assessment Problem 6 develop the full intent of 6.NS.6a (Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite). Mid-Unit Problem 6 states, “Point P is shown on the number line below.” (P labeled at -2.5) “a. What is the opposite of the value of Point P? B. What is the value of -(-P)?” Post-Unit Problem 6 states, “Use the number line below to answer the questions that follow. a. Locate – (−2) on the number line and label it P. b. Is the opposite of 2 located at the same place as point P on the number line? Explain why or why not.”

• In Unit 6, Equations and Inequalities, Post-Unit Assessment problems support the full development of MP2 (Reason abstractly and quantitatively), as students decontextualize a situation to represent it using variables and symbols and then recontextualize in order to interpret what their answer means in regard to the situation at hand. Problem 2 states, “A shelf has four books on it. The weight, in pounds, of each of the four books on the shelf is listed below. 2.5, 3.2, 2.7, 2.3. Which inequality represents the weight, 𝑤, of any book chosen from the shelf? a. 𝑤 > 2.3 b. 𝑤 < 2.4 c. 𝑤 > 3.2 d. 𝑤 < 3.3.” Problem 10 states, “Carlos has at most $40 to spend on food for a barbeque. He wants to buy hot dogs and hamburgers and then rolls for each. Carlos has already spent $31.50 on the hot dogs and hamburgers. Let 𝑟 represent the amount of money that Carlos can spend on the rolls. Write and solve an inequality. Explain what your solution means in context of the situation.”

• In Unit 7, Geometry, Post-Unit Assessment problems support the full development of MP3 as students construct a viable argument about the area of a triangle on a coordinate grid and critique the reasoning of others. Problem 5 states, “Part A: What is the area of triangle A in square units? [The triangle shown has points at (1,2) (4,2) and (7,6)]. Part B: Your teacher tells you to draw Triangle B with coordinate points at (1, 2), (4, 2), and (4, 6). A student in class claims that the area of Triangle B will be less than the area of Triangle A. Do you agree with the student? Explain why or why not.”

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

There are opportunities for students to investigate grade-level mathematics at a higher level of complexity. Often, “Challenge” is written within a Problem Set or Anchor Problem Guiding Questions/ Notes to identify these extensions. According to Math Teacher Tools, Preparing to Teach Fishtank Math, Components of a Math Lesson, “Each Anchor Task/Anchor Problem is followed by a set of Guiding Questions. The Guiding Questions can serve different purposes, including: scaffolding the problem, more deeply engaging students in the content of the problem, and extending on the problem. Not all Anchor Tasks/Problems include Guiding Questions that cover all three purposes. Also, not all Guiding Questions are meant to be asked to the whole class; rather, it should be at the discretion of the teacher to determine how, when, and which questions should be used with which students.” As such, teachers determine how, when, or which students might engage with higher levels of complexity. Examples include:

• In Unit 1, Understanding and Representing Ratios, Lesson 3, Problem Set, Open Middle Finding Equivalent Ratios, contains “Challenge!” after the link, indicating this is a challenging activity for students. 

• In Unit 2, Unit Rates and Percent, Lesson 2, Anchor Problem 2, students are given an unfamiliar context to reason about rates. The materials state, “Hippos sometimes get to eat pumpkins as a special treat. a. If 3 hippos eat 5 pumpkins, how many pumpkins per hippo is that? b. If there are 9 hippos, how many pumpkins do you need? c. If 3 hippos eat 5 pumpkins, how many hippos per pumpkin is that?  d. If you have 30 pumpkins, how many hippos can you feed? Guiding questions: What is the ratio of hippos to pumpkins? What is the ratio of pumpkins to hippos? What are two rates that you’re able to write from this situation?  How can you use the rates to answer parts b and d? What other strategies do you know to answer parts b and d? Notes: This Anchor Problem uses an unfamiliar context to get students to reason about the units at hand, rather than relying on intuition from a familiar context like miles per hour (MP.2). Students find both rates associated with the ratio of hippos to pumpkins and they see how each rate can be useful in answering different questions. However, if students struggle to make sense of the different rates given the unfamiliar context, they can write analogous problems using a familiar context first (MP.1). Ensure students return to the given context after they gain more understanding.” (6.RP.A.2)

• In Unit 5, Numerical and Algebraic Expressions, Lesson 4, Problem Set states, “Challenge: Explore the Pythagorean Theorem and Fermat's Last Theorem. Can you find values for a, b, and c, such that $$a^2+b^2=c^2$$? What about $$a^3+b^3=c^3$$?”

The materials reviewed for Fishtank Plus Math Grade 6 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, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1 uses a handout with ratio shapes, allowing students to cut out and sort/group shapes as they describe ratios for groups of objects. The materials state, “This Anchor Problem works best if the shapes are cut out and can be physically moved around into different groups, either by the teacher or by the students (MP.5). See ratio shapes handout for a copy of the shapes.”

• In Unit 3, Multi-Digit Fraction Computation, Lesson 14, Anchor Problem 3 and Problem Set Guidance both use domino cards and a “Factor Game” to teach and reinforce prime factorization. The game is connected to written methods as students transition to writing numbers as a product of prime factors. The materials state, “This problem requires some materials preparation in order to play the factor dominoes game. These dominoes cards include pictures for numbers 1–60 and can be printed so the pictures can be cut out and optionally pasted onto a cardstock or different colored paper background.”

• In Unit 8, Statistics, Lesson 3, Problem Set Guidance, includes a link to “StatKey Descriptive Statistics for One Quantitative Variable — (This website includes several real-world data sets and their histograms. Select one Quantitative Variable, the data topics, and click on the histogram tab).”