7th Grade - Gateway 3

The materials reviewed for Fishtank Plus Math Grade 7 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 4, Equations and Inequalities, 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 7th Grade Math, Unit 4 Equations and Inequalities. 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 7.

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, Proportional Relationships, Lesson 4, Tips of Teachers provide guidance for teachers to support students in connecting proportional relationships from a table to an equation. The materials state, “Lessons 4 and 5 focus on representing proportional relationships as equations. Equations are abstract and can be challenging for some students to grasp. Encourage students to return to the table to show the relationship between the two quantities, either adding a column to show the constant of proportionality or drawing an arrow across rows and indicating the multiplication. Ensure that students know what the variables in the equation represent to keep the context connected to the abstract form.”

• In Unit 4, Equations and Inequalities, Lesson 10, Anchor Problem 2 notes include guidance for teachers to address common misconceptions with regards to inequality symbols. The materials state, “This is another approach to expose students to inequalities with negative coefficients. Rather than simply telling students to ‘flip the symbol’ when they divide or multiply by a negative number, this problem guides them through determining the direction of the inequality by reasoning about solutions. The solution to the corresponding equation tells students where in the number line to start with their ray, and testing a value will tell them the direction in which to find all solutions.”

• In Unit 7, Statistics, Lesson 4, Anchor Problem 3 Notes provide guidance for teachers to review key concepts about data sets with students. The materials state, “Use this Anchor Problem to review the difference between a measure of center and a measure of variability. In the next lesson, students will learn the mean absolute deviation (MAD) as another measure of variability.”

The materials reviewed for Fishtank Plus Math Grade 7 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, Operations With Rational Numbers, Unit Summary, Intellectual Prep Unit Launch, Content 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. “Looking ahead, students continue their study of the number system in 8th grade with an introduction to irrational numbers. In studying 8.NS.A, students learn that all rational numbers have a terminating or repeating decimal expansion, and those that do not are considered irrational. Students use their knowledge of rational numbers to estimate and reason about the values of irrational numbers. The problems below are examples of problems from the 8th grade curriculum. Notice how students build upon their prior knowledge of rational numbers from 6th and 7th grades. Students’ abilities to compute with rational numbers will be needed throughout their algebra work in middle school and beyond. Simplifying expressions and solving equations will require students to use both properties of operations and their skills with adding, subtracting, multiplying, and dividing signed rational numbers. The problem below is an example from 8th grade where students solve for the variable. Take note of the skills students need with rational number computation in order to solve. In high school, students will learn additional properties of rational and irrational numbers (HSN.RN.B), and they will learn how computing with polynomials is analogous to computing with rational numbers in that they form a closed system under the four operations (HSA.APR).”

• In Unit 2, Operations with Rational Numbers, Lesson 12, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “There are several different ways to introduce multiplication of signed numbers. This lesson focuses in on two approaches: using a number line (Anchor Problem #1) and using properties of operations (Anchor Problem #2). The following links offer additional great insight into this concept and other approaches: Why is a Negative Times a Negative Positive? by SERP Poster Problems (Great explanations based on patterns). Why Is a Negative Times a Negative a Positive? from the blog Mathematical Musings. Why a negative times a negative makes sense by Kahn Academy. Inquiry into positive and negative integer rules from the blog The Reflective Educator.”

• In Unit 4, Equations and Inequalities, Lesson 10, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “In Lesson 10, students encounter inequalities with negative coefficients and must make sense of an unexpected solution. The three Anchor Problems guide students through different approaches to understanding these problems conceptually rather than as a trick of “flipping the inequality symbol.” A common misconception that happens if students learn only the rule and not the conceptual understanding behind it is to over-apply the rule whenever there is a negative in the problem. For example, a student who does not understand this concept may solve 3x > −9  and −3x ﹥ 9 the same way as x ﹤ −3. If the student were to test out possible values for the solution into the inequality, the student would see that this is the solution to only one of those inequalities. 

• In Unit 5, Percent and Scaling, Unit Summary, Intellectual Prep Unit Launch, Content 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. “When students study statistics and probability in 7th and 8th grades, they are likely to see and use percentages. In statistics, students may encounter data or graphs that are represented with percentages. In studying probability, students may encounter situations with percent chances or likelihoods. Knowing what percentages mean and how to work with them as numerical values, will allow students to focus on and make greater sense of the new content they are learning. Two examples are shown below, one aligned to 7th grade standard 7.SP.C.7, and the other aligned to 8th grade standard 8.SP.A.4. In the 8th grade problem, students see how finding relative frequency in percent form can provide more accurate and less misleading information than numerical data values alone, similar to percent error in measurement. Later in high school, when students study compound interest and exponential expressions, they may think back and recall how percentages were used in simple interest calculations in middle school. The problem below is from Algebra 2 aligned to standard A.SSE.B.3c.”

The materials reviewed for Fishtank Plus Math Grade 7 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 7th 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 7th Grade Math, Unit 5, Percent and Scaling, 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 6, Geometry, Lesson 18, the Core Standard is identified as 7.G.B.6. The Foundational Standard is identified as 6.G.A.4. 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, Operations with Rational Numbers, Unit Launch, Connections states, “The cluster heading for 7.NS.A says to ‘Apply and extend previous understandings of operations with fractions’. In line with this, students have a lot of prior knowledge around computing with fractions that they bring with them to this unit. Furthermore, students have prior knowledge around rational numbers as they exist on the number line, both to the right and the left of zero. In this 5th grade problem aligned to 5.NF.A.1, students add mixed numbers with different denominators. They are first asked to find an estimate before computing. This type of reasoning will be helpful to students in 7th grade when they work with more complex values, both positive and negative. Looking ahead, students continue their study of the number system in 8th grade with an introduction to irrational numbers. In studying 8.NS.A, students learn that all rational numbers have a terminating or repeating decimal expansion, and those that do not are considered irrational. Students use their knowledge of rational numbers to estimate and reason about the values of irrational numbers.”

• In Unit 4, Equations and Inequalities, Unit Launch, Connections states, “In 6th grade, students spend time understanding what it means to be a solution to an equation or an inequality (6.EE.B.5). They learn that they can use substitution to determine if a specific value is a solution, based on whether or not they reach a true statement. They compare equations and inequalities and discover that inequalities have infinite solutions, compared to the unique solutions of equations. These concepts are important for students to understand as they delve into the work of solving. In 8th grade, positive and negative rational numbers appear regularly throughout students' work, as seen in the examples above. While all of the equations above have unique solutions, students will also encounter equations that lead to results such as 5=3 or 8=8, indicating the equation has no solution or infinite solutions, respectively.”

• In Unit 7, Statistics, Unit Summary includes an overview of how the content in sixth grade builds from previous grades. “In sixth grade, students began their study of statistics by understanding what makes a statistical question. They studied shapes of distributions of data and calculated measures of center and spread. Students made connections between the data and the contexts they represented, ensuring the numerical aspects of statistics were not separated from the statistical question that drove the analysis. All of these understandings will support seventh-grade students in their work in this unit.”

The materials reviewed for Fishtank Plus Math Grade 7 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. 7th Grade, 7.EE.4a, 7.EE.3, 7.NS.1, 7.NS.2.”

• In Math Teacher Tools, Academic Discourse, Tiers of Academic Discourse, Overview, “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 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 7th Grade Course Summary, Course Material Overview, Course Material List 7th Grade Mathematics states, “The list below includes the materials used in the 7th 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, and 7, one per student. 

• Graph paper is used in Units 1, 2, 5, and 6, one ream. 

• Rulers are used in Unit 1, 5, and 6, one per student.

• String or flexible measuring tools are used in Units 5 and 6, one per student. 

• Unit 6 uses protractors, one per student.

• Counters are used in Unit 8, forty to sixty, 11 per small group.

• In Unit 2, Operations with Rational Numbers, Lesson 4, students model the addition of integers using a number line (7.NS.A.1.B and 7.NS.A.1.D). Tips for Teachers states, “The following materials may be used in this lesson: number line and game piece from Lesson 1.” Anchor Problem 1 states, “The number line below represents the road that Joshua lives on, with his home located at point 0. The numbers on the number line represent the number of miles from Joshua’s house, either east or west. Throughout the week, Joshua goes on trips and errands along this road.For each day described in the chart, model Joshua’s trip and determine where on the number line he ends up each day. Write an addition equation to represent it.” (several scenarios are listed). Notes state, “Students can model each trip physically, by using their number lines and game pieces, or visually, by drawing arrows on number lines. The focus of this problem is for students to make a connection between positive and negative numbers and how those can be modeled as actions on the number line when they are combined.”

• Brown paper bags are used in Unit 8, 10-12 (two per pair of students). In Unit 8, Probability, Lesson 2, Tips for Teachers states, “This lesson requires some prior preparation and materials (brown paper bags and different colored cubes) for Anchor Problem #1. See the notes in Anchor Problem #1 for further information.”

The materials reviewed for Fishtank Plus Math Grade 7 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, Proportional Relationships, Unit Summary, Unit Assessment Answer Key denotes standards addressed for each question. Question 2 is aligned to 7.RP.1 and states, “A vehicle uses 1\frac{1}{8} gallons of gasoline to travel 13\frac{1}{2} miles. At this rate, how many miles can the vehicle travel per gallon of gasoline? a. \frac{16}{242} b. \frac{4}{3} c. 12 d. 13.”

• In Unit 2, Operations with Rational Numbers, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 1 is aligned to 7.NS.1a and states, “Luke has a savings account with an initial balance of $380. He makes two transactions. After the two transactions, the balance of Luke’s savings account is the same as the initial balance. Which of the following could be the two transactions that Luke made? a. A withdrawal of $200 and a deposit of $180 b. A withdrawal of $215 and a deposit of $215 c. A withdrawal of $200 and a withdrawal of $180 d. A deposit of $400 and a withdrawal of $20.”

• In Unit 4, Equations and Inequalities, Unit Summary, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 4 is aligned to MP6 and states, “Solve the equation, showing all of your work. 7n + 14 = 21.” 

• In Unit 5, Percent and Scaling, Unit Assessment, Answer Key includes a constructive response and 3-point rubric with the aligned grade-level standard. Question 5 is aligned to 7.EE.3 and states, “A museum opened at 8:00am. In the first hour, 350 people purchased admission tickets. In the second hour, 20% more people purchased admission tickets than in the first hour. Each admission ticket cost $17.50. What was the total amount of money paid for all the tickets purchased in the first two hours?”

• In Unit 7, Statistics, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 2 is aligned to MP4 and states, “Lorenzo wants to know how long it takes students to get to his school in the morning. He doesn’t have enough time to ask every student. Describe in detail a strategy he could use to get the information he needs.”

The materials reviewed for Fishtank Plus Math Grade 7 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, “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 1, Proportional Relationships, Lesson 2, Target Task, students represent proportional relationships in tables, and determine the constant of proportionality. The materials state, “A family took a road trip down the East coast. On average, they traveled at a constant speed, represented in miles and hours in the table below. a. Describe the relationship between hours and miles. b. How fast is the family driving? c. What is the constant of proportionality? Explain what it means in this situation.” The Mastery Response states, “a) For every hour spent driving, the family covered 60 miles. b) 60 miles per hour. c) The constant of proportionality is 60. In context, this describes how many miles the family covers in 1 hour, or their speed in miles per hour.” 

• In Unit 2, Operations with Rational Numbers, Post-Unit Assessment Analysis, Problem 3 states, “Which expression is equivalent to 4 − (−7)? a. 7 + 4 b. 4 − 7 c. −7 − 4 d. −4 + 7. Commentary: This problem assesses students’ understanding of subtraction as addition of the additive inverse. Students must select an expression that is equivalent; the expectation is that students use their conceptual understanding to re-write the expression, not that students compute the value of each expression and select the equivalent value. It may be valuable to see how students perform on related problem #13.”

• In Unit 6, Geometry, Mid-Unit Assessment, Answer Key, Problem 7 states, “Pablo bought a new circular rug for his bedroom. From the center to the edge of the rug is 1.5𝑚. a. How much floor space does the rug cover? b. The lease for Pablo’s apartment says that he has to cover at least 80% of the floor of each room with rugs. If Pablo’s bedroom is a square with 3.1𝑚 long sides, is his new rug large enough to meet the lease’s requirement? Justify your answer.” The Answer Key provides the correct answers and a 3-point rubric for teachers to follow. It states, “3 points. Student response demonstrates an exemplary understanding of the concepts in the task. The student points correctly and completely answers all aspects of the prompt. 2 points. Student response demonstrates a good understanding of the concepts in the task. The student arrived at an acceptable conclusion, showing evidence of understanding of the task, but some aspect of the response is flawed. 1 point. Student response demonstrates a minimal understanding of the concepts in the task. The student arrived at an incomplete or incorrect conclusion, showing little evidence of understanding of the task, with most aspects of the task not completed correctly or containing significant errors or omissions. 0 points. Student response contains insufficient evidence of an understanding of the concepts in the task. Work points may be incorrect, unrelated, illogical, or a correct solution obtained by chance.”

• In Unit 8, Probability, Pre-Unit Assessment, Teacher Answer Key & Guide, Problem 4 states, “Tony, owner of Tony’s Pizza Parlor, is expanding his business. Right now, his pizza parlor uses 8lbs of pepperoni for every 3lbs of green peppers. If the expanded business has the same relationship between ingredients, and Tony thinks he’ll need 19.5lbs of green peppers, how many pounds of pepperoni will he need? Potential Course of Action, If needed, review this idea before Lesson 4, where students use probabilities to determine long run frequencies. For example, include problems where students find missing values in proportional relationships in a homework for Lesson 3 or a warm up for Lesson 3. Find problems and other resources in these Fishtank lessons: Grade 6, Unit 1, Lesson 6, 11, 14. Grade 7, Unit 1, Lesson 1, 3, 15.”

The materials reviewed for Fishtank Plus Math Grade 7 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, Operations with Rational Numbers, Post-Unit Assessment, Problems 8, 10, and 14 develop the full intent of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers). Problem 8 states, “A baker regularly uses flour to make pastries for the bakery. At the beginning of the month, the baker has a 25 pound-bag of flour. Each week she uses $$5\frac{3}{5}$$ pounds of flour. a. Write an expression to represent the change in weight of the bag of flour over 4 weeks. b. How much flour is left in the bag after 4 weeks? Show your work clearly.” Problem 10 states, “The table below shows the weekly change in the price of one gram of gold for four weeks. a. By how much did the price of one gram of gold change from the beginning of week 1 to the end of week 4? Did the price increase or decrease? Show your work clearly. B. At the end of week 4, the price per gram of gold was $39.28. What was the price per gram of gold at the beginning of week 1?” Problem 14 states, “Ava and Jiao each swam a two-lap swimming race. Ava took 31.49 seconds to finish her first lap and 30.03 seconds to finish her second lap. Jiao finished her two-lap swimming race 1.76 seconds faster than Ava. What was Jiao’s total swimming time, in seconds, after she finished her two-lap race?”

• In Unit 4, Equations and Inequalities, Mid-Unit Assessment problems support the full development of MP6 (Attend to precision) as students solve equations with rational numbers and look for and describe errors within sample solutions. The materials state, “For numbers 1-4, solve the equation, showing all work. 1.) $$\frac{1}{3}+x=\frac{1}{6}$$, 2.) 0.4a + 1.2 = 3.6, 3.) $$\frac{1}{2}$$(n - 6) = 44, 4.) -8q + 16 = -16.” Problem 5 states, “Ms. Jones asked students to solve the equation 10 − 2𝑥 = −18. Allen shows the following work: a. Identify and describe Allen’s error. b. Solve 10 − 2𝑥 = −18.” 

• In Unit 5, Percent and Scaling, Post-Unit Assessment problems support the full intent of MP4 as students reason about equivalent expressions and model with mathematics. Problem 1 states, “A store owner uses the expression below to calculate the sale price of each CD player he is discounting. 𝑟 − 0.2𝑟 In the expression, 𝑟 represents the regular price of a CD player. Which of the following is equivalent to the expression? a. 0.1𝑟 b. 0.8𝑟 c. 0.9𝑟 d. 1.2r.” Problem 2 states, “Jordan’s dog weighs 𝑝 pounds. Emmett’s dog weighs 25% more than Jordan’s dog. Which expressions represent the weight, in pounds, of Emmett’s dog? Select two correct answers. a. 0.25𝑝 b. 1.25𝑝 c. 𝑝 + 0.25 d. 𝑝 + 1.25 e. 𝑝 + 0.25𝑝.”

• In Unit 8, Probability, Post-Unit Assessment, Problem 5 develops the full intent of 7.SP.8c (Design and use a simulation to generate frequencies for compound events). The materials state, “Martina read that approximately 10% of all people are left-handed. She wants to design a simulation that she can use to model this situation. Part A: In the simulation, Martina has a spinner with sections of equal size. One section is labeled ‘L’ (left) and the rest of the sections are labeled ‘R’ (right). For this simulation to be as accurate as possible, what is the total number of sections that the spinner should have? Part B: Martina wants to use her simulation to approximate the probability of selecting exactly 2 right-handed people when 3 people are randomly selected. Martina spins the arrow on the spinner 3 times and records the resulting letters. She performs the simulation 30 times. The results of the simulation are shown. Based on the results of this simulation, when 3 people are randomly selected, exactly 2 right-handed people are selected approximately ____ percent of the time. What number best fills in the blank? a. 10 b. 15 c. 20 d. 25.”

The materials reviewed for Fishtank Plus Math Grade 7 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, Proportional Relationships, Lesson 7, Anchor Problem 3, Guiding Questions state, “Challenge: Suppose the independent and dependent variables in this problem are switched. How does the problem change? Is it still a proportional relationship? What does the point (1, r) represent in this new situation?”

• In Unit 3, Numerical and Algebraic Expressions, Lesson 6, Problem Set states, “Challenge: Expression A is given by 3−x. Expression B  is given by −x−3. Write a simplified expression that represents  −A−B−B−A.” 

• In Unit 7, Statistics, Lesson 6, Problem Set states, “Illustrative Mathematics Valentine Marbles — Challenge: This is a lengthy problem that would be good for students to work on in pairs.”

The materials reviewed for Fishtank Plus Math Grade 7 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, Proportional Relationships, Lesson 18, Problem Set Guidance includes a link to “Desmos Tile Pile.” Students work with this virtual manipulative to reinforce proportional reasoning in solving problems.

• In Unit 5, Percent and Scaling, Lesson 4, Problem Set Guidance includes a link to “Andrew Stadel's 3-Act Math Tasks iPad Usage.” This virtual manipulative is used to reinforce finding percentages given a part and the whole.

• In Unit 6, Geometry, Lesson 6, Anchor Problem 1, students use a virtual manipulative, “GeoGebra, Circumference of a Circle (Drag the center dot to “unroll” the circle, and drag the blue dot to change the diameter.)” to explore the relationship between the diameter and circumference of a circle.