## Fishtank Plus Math

##### v1.5
###### Usability
Our Review Process

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### Overall Summary

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for Usability: meet expectations for Teacher Supports (Criterion 1), meet expectations for Assessment (Criterion 2), and partially meet expectations for Student Supports (Criterion 3).

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, each grade’s materials are coherent and consistent with the CCSSM.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

##### Indicator {{'1a' | indicatorName}}

Materials assess the grade-level content and, if applicable, content from earlier grades.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into seven units and each unit contains a Pre-Unit Assessment, Mid-Unit Assessment, and Post-Unit Assessment. Pre-Unit assessments may be used “before the start of a unit, either as part of class or for homework.” Mid-Unit assessments are “designed to assess students on content covered in approximately the first half of the unit” and may also be used as homework. Post-Unit assessments “are designed to assess students’ full range of understanding of content covered throughout the whole unit.” Examples of Post-Unit Assessments include:

• In Unit 2, Multiplication and Division of Whole Numbers, Post-Unit Assessment, Problem 3 states, “Which expression matches the statement, ‘the sum of 2 and 4 subtracted from 9’? A. 2 + 9 – 4; B. 9 – 2 + 4; C. 9 – (2 + 4); D. (2 + 4) – 9.” (5.OA.2)

• In Unit 5, Multiplication and Division of Fractions, Post-Unit Assessment, Problem 4 states, “Jin had 60 stickers in her collection. She gave $$\frac{3}{5}$$ of the stickers to her friend. How many stickers did Jin give to her friend? A. 12; B. 20; C. 36; D. 40.” (5.NF.4)

• In Unit 6, Multiplication and Division of Decimals, Post-Unit Assessment, Problem 6 states, “a. If 358 × 25 = 8,950, then what is 3.58 × 25 equal to? Explain your reasoning. b. If 12 ÷ 2 = 6, then what is 120 ÷ 0.2 equal to? Explain your reasoning.” (5.NBT.7)

• In Unit 7, Patterns and Coordinate Plane, Post-Unit Assessment, Problem 1 states, “Which of the following correctly describes a way to plot the point (2, 5) on a coordinate plane? A. Start at the origin. Move 2 units up the 𝑦-axis, then move 5 units to the right. Plot the point there.; B. Start at the top of the 𝑦-axis. Move 2 units down the 𝑦-axis, and then move 5 units to the right. Plot the point there.; C. Start at the origin. Move 2 units to the right on the 𝑥-axis, and then move 5 units up. Plot the point there.; D. Start at the top of the 𝑦-axis. Move 2 units to the right, and then move 5 units down. Plot the point there.” (5.G.1)

##### Indicator {{'1b' | indicatorName}}

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The instructional materials provide extensive work in Grade 4 by providing Anchor Tasks, Problem Sets, Homework, and Target Tasks for each lesson. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 19, Target Task engages students in extensive work in 4.OA.3 (solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). It states, “In one year a factory used 11650 meters of cotton, 4,950 fewer meters of silk than cotton, and 3,500 more meters of wool than silk. 1. How many meters in all were used in the three fabrics? Show or explain your work. 2. Assess the reasonableness of your answer.”

• In Unit 2, Multi-Digit Multiplication, Lesson 19, Problem Set, Problem 2 engages students in extensive work in 4.NBT.5 (multiply a whole number of up to four digits by a one-digit whole number, and multiply two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). It states, “2. Use the same method as Felicia to complete an area model and an equation to solve each of the following multiplication problems. a. 14 × 22 b. 25 × 32.”

• In Unit 6, Fraction Operations, Lesson 12, Homework, Problems 1-3 engage students in extensive work in 4.NF.3c (add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction). It states, “1. Solve. Show or explain your work. a. 4\frac{1}{3}+\frac{1}{3} b. 4\frac{1}{4}+\frac{2}{4} c. \frac{2}{6}+3\frac{4}{6} d. \frac{5}{8}+7\frac{3}{8} 2. Find the sum in two ways. 5\frac{7}{10}+\frac{4}{10}; 3. Georgia was solving #2, and wrote 5\frac{11}{10} as her answer. How would you suggest Georgia change the way she has recorded her answer? Why is it helpful to record answers that way?”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 4 standards through a consistent lesson structure, including Anchor Tasks, Problems Sets, Homework Problems, and Target Tasks. Anchor Problems include a connection to prior knowledge, multiple entry points to new learning, and guided instruction support. Problem Set Problems engage all students in practice that connects to the objective of each lesson. Target Task Problems can be used as formative assessment. Each unit is further divided into topics. The lessons within each topic build on each other, meeting the full intent of the standards. Examples of where the materials meet the full intent include:

• In Unit 3, Multi-Digit Division, Lesson 9 provides an opportunity for students to engage with the full intent of standard 4.NBT.6 (find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models).  Target Task, Problem 2 is written and solved using the standard algorithm. It states, “Here is a calculation of 8,472 ÷ 5. a. There’s a 5 under the 8 in the 8,472. What does this 5 represent? b. What does the subtraction of 5 from 8 mean? c. Why is a 4 written next to the 3 from 8-5?” Problem set, Problem 2 states, “Write a division problem whose quotient is 314 R 7. Explain how you came up with it.”  Homework, Problem 4 states, “Tamieka is making bracelets. She has 3,467 beads. It takes 8 beads to make each bracelet. How many bracelets can she make? How many more beads would she need to be able to make another bracelet?”

• In Unit 4, Shapes and Angles, Topic B: Measures of Angles, Lessons 5-10 provide an opportunity for students to engage with the full intent of standards 4.MD.5 (recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement) and 4.MD.6 (measure angles in whole-number degrees using a protractor. Sketch angles of specified measure). In Lesson 7, Problem Set, Problem 1 states, “Do not use your protractor to solve! a. Look at the angle shown. Which measure is closest to the measure of the angle? a. 140 b. 90 c. 40. d. 15.” In Lesson 8, Problem Set, Problem 2 states, “Maria says that this angle measures 153 degrees. Is she correct or incorrect? Why.” Lesson 9, Anchor Tasks, Problem 2 states, “Sketch angles that have each of the following angle measures. a. 80° b. 133°”

• In Unit 5, Fraction Equivalence and Ordering, Topic B: Comparing and Ordering Fractions,  Lessons 7-11, provides an opportunity for students to engage with the full intent of 4.NF.2 (compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numberators, or by comparing to a benchmark fraction such as \frac{1}{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >,=, or <, and justify the conclusions, e.g., by using a visual fraction model).  In Lesson 7, Problem Set, Problem 2 states, “Compare each pair of fractions using >, <, or =. a. \frac{3}{4} ___ \frac{3}{7} b. \frac{2}{5} ___ \frac{4}{9} c. \frac{2}{3} ___ \frac{5}{6} d. \frac{3}{8} ___ \frac{1}{4} e. \frac{7}{11} ___ \frac{7}{13} f. \frac{8}{9} ___ \frac{2}{3} g. \frac{2}{3} ___ \frac{5}{6} h. \frac{3}{4} ___ \frac{7}{12}.” Problem 5 states, “Select True if the comparison is true. Select False if the comparison is not true. \frac{89}{100}>\frac{9}{10}; \frac{7}{12}<\frac{2}{3}\frac{3}{5}>\frac{4}{10}.” In Lesson 9, Homework, Problem 3 states, “Rowan has 3 pieces of yarn, as described below. A red piece of yarn that is \frac{3}{4} foot long; A yellow piece of yarn that is \frac{6}{8} foot long; A blue piece of yarn that is \frac{4}{12} foot long. Which number sentence correctly compares the lengths of 2 of these pieces of yarn? A. \frac{3}{4}<\frac{6}{8}; B. \frac{4}{12}<\frac{3}{4}; C. \frac{3}{4}>\frac{6}{8}; D. \frac{4}{12}>\frac{6}{8}.”

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.

##### Indicator {{'1c' | indicatorName}}

When implemented as designed, the majority of the materials address the major clusters of each grade.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major work of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade:

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 5.1 out of 7,  approximately 73%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 99 out of 129, approximately 77%. The total number of lessons includes 122 lessons plus 7 assessments or a total of 129 lessons.

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work), is 109 out of 141, approximately 77%. There are a total of 19 flex days and 15 of those days are included within units focused on major work, including assessments. By adding 15 flex days focused on major work to the 94 lessons devoted to major work, there is a total of 109 days devoted to major work.

• The number of days devoted to major work (excluding flex days, while including assessments and supporting work connected to the major work) is 99 out of 129, approximately 77%. While it is recommended that flex days be used to support major work of the grade within the program, there is no specific guidance for the use of these days.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 77% of the instructional materials focus on major work of the grade.

##### Indicator {{'1d' | indicatorName}}

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 5 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. These connections are sometimes listed for teachers as “Foundational Standards'' on the lesson page. Examples of connections include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 20, Homework, Problem 3 connects the supporting work of 5.OA.1 (use parentheses, brackets, or braces in numerical expression, and evaluate expressions with these symbols) to the major work of 5.NBT.5 (fluently multiply multi-digit whole numbers using the standard algorithm). It states, “Frances is sewing a border around 2 rectangular tablecloths that each measure 9 feet long by 6 feet wide. If it takes her 3 minutes to sew on 1 inch of border, how many minutes will it take her to complete her sewing project? Write an expression, and then solve.”

• In Unit 3, Shapes and Volume, Lesson 5, Anchor Task, Problem 3 connects the supporting work of 5.OA.2 (write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). It states, “Akiko and Philip are finding the volume of the following rectangle prism. Philip says that you have to multiply length by width by height, so you have to multiply 10 × 14 × 2. Akiko says the computation will be easier if you multiply 10 × 2 × 14. a. Is Philip correct? Must the dimensions be multiplied in that order? Show or explain your thinking. b. Why do you think Akiko thinks that multiplying 10 × 2 × 14 will be an easier computation? Is it possible to multiply the dimensions in that order? Show or explain your thinking. c. Use what you’ve concluded from Parts (a) and (b) to explain how you would calculate the volume of a rectangular prism whose length is 4 feet, width is 7 feet, and height is 15 feet.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 16, Problem Set, Problem 1 connects the supporting work of 5.OA.2 (write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.NF.4 (apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction). It states, “The total distance around a running track is $$1\frac{5}{8}$$ miles. Wayne ran $$\frac{1}{4}$$ of the track. Which of the following equations can be used to find d, the distance in miles that Wayne ran? a. $$\frac{1}{4}×\frac{13}{8}$$, b. $$\frac{1}{4}×\frac{13}{8}$$, c. $$\frac{4}{1}×\frac{13}{8}$$, d. $$\frac{4}{1}×\frac{15}{8}$$”.

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Target Task, Problem 2  connects the supporting work of 5.MD.1 (convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-stop, real world problems) with the major work of 5.NF.4 (apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). It states, “Solve. Show or explain your work. Yolanda took a bus to visit her grandmother. She brought a CD to listen to on the bus. The CD is 78 minutes long. The bus ride was $$2\frac{1}{2}$$ hours long. How many minutes longer was the bus ride than the CD?”

##### Indicator {{'1e' | indicatorName}}

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples of connections from supporting work to supporting work and/or from major work to major work throughout the grade-level materials, when appropriate, include:

• In Unit 1, Place Value with Decimals, Lesson 9, Target Task, Problem 1 connects the major work of 5.NBT.A to the major work of 5.NF.B as students write numbers with decimal values in expanded form and use whole numbers multiplied by the fractional value of the place (e.g., 4 x 110).  It states, “A number is given below. 136.25 In a different number, the 6 represents \frac{1}{10} of the value of the 6 in the number above. What value is represented by the 6 in the other number? A. Six hundredths B. Six tenths C. Six ones D. Six tens.”

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 4, Anchor Tasks, Problem 2 connects the major work of 5.NBT.A to the major work on 5.NBT.B as students apply their understanding of the value of a digit in multiplying numbers. It states, “Solve.

1. 60 × 5 = ___

2. 60 × 50 = ___

3. 60 × 500 = ___

4. 60 × 5,000 = ___.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 13, Anchor Tasks, Problem 2 connects the major work of 5.NBT.B to the major work of 5.NBT.A, as students perform operations with multi-digit whole numbers and decimals to hundredths, and understand the place value system. It states, “Solve. Show your work with an area model. a. 0.3 + 0.5 b. 0.64 + .07.”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Tasks connects the supporting work of 5.G.A to the supporting work 5.OA.B as students represent real world problems within the first quadrant of a coordinate plane, and generate patterns using rules. It states, “Jessica has $15 saved. She earns$6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs 45. How many hours does Jessica need to babysit in order to be able to buy it?” • In Unit 7, Patterns and the Coordinate Plane, Lesson 12, Homework, Problem 5 connects the supporting work of 5.MD.B to the supporting work of 5.G.A as students enter data in a table, interpret the data, and complete a line graph on a coordinate plane. It states, “5. Use the following two patterns to complete Parts (a)—(d): Pattern for x-coordinates: Start at 1, add 4; Pattern for y-coordinates: Start at 3, add 4; A. Complete the following table. B. Plot each point on the coordinate plane to the right. C. Use a straightedge to construct a line through these points. D. Give the coordinates of two other points that fall on this line with the x-coordinates greater than 25. (___, ___ ) and (___, ___) E. How did you find two other points that would lie on this line?” ##### Indicator {{'1f' | indicatorName}} Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Content from future grades is identified within materials and related to grade-level work. These references are consistently included within the Unit Summary. Examples include: • In Unit 3, Shapes and Volume, Unit Summary states, “In Grade 6, students will explore concepts of length, area, and volume with more complex figures, such as finding the area of right triangles or finding the volume of right rectangular prisms with non-whole-number measurements (6.G.1, 6.G.2). Students will even rely on their understanding of shapes and their attributes to prove various geometric theorems in high school (GEO.G-CO.9—11). Thus, this unit provides a nice foundation for connections in many grades to come.” • In Unit 4, Addition and Subtraction of Fractions/Decimals, Unit Summary states, “As previously mentioned, students will explore the other operations, multiplication and division, of fractions and decimals in Units 5 and 6, including all cases of fraction and decimal multiplication and division of a unit fraction by a whole number and a whole number by a unit fraction (5.NF.3–7, 5.NBT.7). In Grade 6, students encounter the remaining cases of fraction division (6.NS.1) and solidify fluency with all decimal operations (6.NS.3). Students then rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals.)” • In Unit 5, Multiplication and Division of Fractions, Unit Summary states, “In Grade 6, students encounter the remaining cases of fraction division (6.NS.1). Work with fractions and multiplication is a building block for work with ratios. In Grades 6 and 7, students use their understanding of wholes and parts to reason about ratios of two quantities, making and analyzing tables of equivalent ratios, and graphing pairs from these tables in the coordinate plane. These tables and graphs represent proportional relationships, which students see as functions in Grade 8” (NF Progression, p. 20). Students will further rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals. • In Unit 7, Patterns and The Coordinate Plane, Unit Summary states, “This work is an important part of ‘the progression that leads toward middle-school algebra’ (6—7.RP, 6—8.EE, 8.F) (K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics, p. 7). This then deeply informs students’ work in all high school courses. Thus, Grade 5 ends with additional cluster content, but that designation should not diminish its importance this year and for years to come.” Materials relate grade-level concepts from Grade 5 explicitly to prior knowledge from earlier grades. These references can be found within materials in the Unit Summary, within Lesson Tips for Teachers, and in the Foundational Skills information in each lesson. Examples include: • In Unit 1, Place Value with Decimals, Unit Summary states, “In Grade 4, students developed the understanding that a digit in any place represents ten times as much as it represents in the place to its right (4.NBT.1). With this deepened understanding of the place value system, students read and wrote multi-digit whole numbers in various forms, compared them, and rounded them (4.NBT.2—3).” • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 5, Foundational Skills lists the standards 4.NBT.B.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm) and 4.NBT.B.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) as foundational standards from Grade 4. It is also noted that these skills are covered in the unit. • In Unit 2, Multiplication and Division of Whole Numbers, Unit Summary states, “In Grade 4, students attained fluency with multi-digit addition and subtraction (4.NBT.4), a necessary skill for computing sums and differences in the standard algorithm for multiplication and division, respectively. Students also multiplied a whole number of up to four digits by a one-digit whole number, as well as two two-digit numbers (4.NBT.5). By the end of Grade 4, students can compute those products using the standard algorithm, but ‘reason repeatedly about the connection between math drawings and written numerical work, help[ing] them come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities’ (Progressions for the CCSSM, ‘Number and Operation in Base Ten, K-5’, p. 14). Students also find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (4.NBT.6). Similar to multiplication, by the end of Grade 4, students can compute these quotients using the standard algorithm alongside other strategies and representations so that the algorithms are meaningful rather than rote.” • In Unit 5, Multiplication and Division of Whole Numbers, Tips for Teachers states, “Students have seen the area model, the partial products algorithm, and the standard algorithm strategies with two-digit by two-digit multiplication in Grade 4.” • In Unit 6, Multiplication and Division of Decimals, Unit Summary states, “In Grade 4, students were first introduced to decimal notation for fractions and reasoned about their size (4.NF.5—7). Then, in the first unit in Grade 5, students developed a deeper understanding of decimals as an extension of our place value system, understanding that the relationships of adjacent units apply to decimal numbers, as well (5.NBT.1), and using that understanding to compare, round, and represent decimals in various forms (5.NBT.2—4). Next, students learned to multiply and divide with whole numbers in Unit 2 (5.NBT.5—6), skills upon which decimal computations will rely.” ##### Indicator {{'1g' | indicatorName}} In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification. The materials reviewed for Fishtank Plus Math Grade 5 foster coherence between grades, materials can be completed within a regular school year with little to no modification. According to the Pacing Guide, “The fifth-grade math curriculum was designed to be implemented over the course of a single school year. It includes seven units of study over 141 instructional days (including days for lessons, flex days, and unit assessments). We intentionally did not account for all 180 instructional days in order for teachers to fit in additional review or extension, teacher-created assessments, and school-based events. Each unit includes a specific number of lessons, a day for assessment, and a recommended number of flex days (see the table below). These flex days can be used at the teacher’s discretion, however, for units that include both major and supporting/ additional work, it is strongly recommended that the flex days be spent on content that aligns with the major work of the grade.” Included in the 141 days are: • 122 lesson days • 12 flex days • 7 unit assessment days There are seven units and, within those units, there are 12 to 24 lessons that contain a mixture of Anchor Tasks, Problem Set Problems, Homework Problems, and Target Tasks. The number of minutes needed to complete each lesson component are aligned to guidance in the Pacing Guide. Each 60 minute lesson is composed of: • 25 - 30 minutes Anchor Tasks • 15 - 20 minutes Problem Set • 5 - 10 minutes Target Task Additionally, the Pacing Guide notes, “it is recommended to also allocate 10 minutes for daily application and 15 minutes for daily fluency. These additional blocks are meant to provide sufficient time and practice for these aspects of rigor.” ###### Overview of Gateway 2 ### Rigor & the Mathematical Practices The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. The materials make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Gateway 2 Meets Expectations #### Criterion 2.1: Rigor and Balance Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately. ##### Indicator {{'2a' | indicatorName}} Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts and provide opportunities for students to independently demonstrate conceptual understanding throughout Grade 5. Materials develop conceptual understanding throughout the grade level. According to Course Summary, Learn More About Fishtank Math, Our Approach, “Procedural Fluency AND Conceptual Understanding: We believe that knowing ‘how’ to solve a problem is not enough; students must also know ‘why’ mathematical procedures and concepts exist.” Each lesson begins with Anchor Tasks and Guiding Questions, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. This is followed by a Problem Set, Homework and Target Task. Examples include: • In Unit 1, Place Value with Decimals, Lesson 2, Problem Set, Problems 4 and 5. Problem 4 states, “Solve. a. 10 × 10 = ____; b. 100 × 1,000 = ____; c. 10 × 100,000 = ____; d. 1,000 × 1,000 = ____; e. 10,000 × 100,000 = ____.” Problem 5 states, “What do you notice about the factors and products in #4?” In addition, the “Discussion of Problem Set,” provides teachers with questions to ask students. For example, “Look at #4e. How did you solve this even though it’s beyond what we’ve done already?” This activity supports conceptual understanding of 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10). • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 12, Anchor Task, Problem 2 states, “Estimate the following quotients. a. 4,212 ÷ 52 b. 1,232 ÷ 28 c. 5,427 ÷ 81.” Guiding Questions include, “What should we round our divisor to? What is a compatible number that we can use in place of the dividend? What is our estimated quotient? Of all three estimates, which one do you think is closest to the actual quotient? Which one do you think is an overestimate? Which one do you think is an underestimate? Why? Why do we round the divisor first, then replace the dividend with a compatible number? What if we rounded the dividend first, then replaced the divisor with a compatible number? Let’s try that with one (or a few) of these computations, then compare it to the actual quotient. For which problems above would rounding both factors to their largest place value give an estimate that we can compute? How does this demonstrate that using compatible numbers is a more reliable strategy to estimate quotients?” This problem allows students to develop, with teacher support, conceptual understanding of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). • In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 4, Anchor Tasks, Problem 1 states, “Solve. a.1 orange + 3 oranges = ____; b. 1 child + 3 adults = ____. 2. What do you notice about #1 above? What do you wonder?” This problem reinforces the importance of comparing like items, as students will need to know/remember to use a common denominator when adding or subtracting fractions. Guiding Questions include, “What do you notice about #1? What do you wonder? Can we use some other ‘unit’ for (b) that would make it possible to add them?” This problem and the accompanying questions guide students to develop conceptual understanding of 5.NF.1 (Add and subtract fractions with unlike denominators). Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Problem Sets and Homework Problems can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding of key concepts, are designed for independent completion. Many of these problems provide opportunities for students to independently demonstrate conceptual understanding. Examples include: • In Unit 3, Shapes and Volume, Lesson 4, Problem Set, Problem 8 states, “A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The student’s work is shown.” The following student work is provided, “Student’s Work - I pack my box full of cubes. Each cube has a volume of 1 cubic inch. I counted 63 cubes in the top layer. Since there are 9 layers of cubes below the top layer, I solved 63 x 9 = 567. So there are 567 cubes. The volume of my box is 567 cubic inches.” Students are then asked the following questions, “a. Explain why the student’s reasoning is incorrect. Provide the correct volume, in cubic inches, of the box. b. A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height in inches, of the second box? Explain or show how you determined the height.” This problem builds conceptual understanding of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). • In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 15, Target Task states, “Lina brought10 to the fair. She spent $2.59 for cotton candy. She spent$3.49 for a toy. How much money did Lina have left?” This activity provides an opportunity for students to demonstrate conceptual understanding of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Problem Set, Problem 1 states, “Solve each problem in two different ways as modeled in the example. You may draw a model to help you. a. \frac{6}{7}×\frac{5}{8}; b. \frac{4}{5}×\frac{5}{8}; c. \frac{2}{3}×\frac{6}{7}; d. \frac{4}{9}×\frac{3}{10}”. These problems allow students to independently demonstrate conceptual understanding of 5.NF.4 (Apply and extend previous knowledge of multiplication to multiply a fraction by a fraction).

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Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for developing procedural skills and fluency while providing opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level.

According to Teacher Tools, Math Teacher Tools, Procedural Skill and Fluency, “In our curriculum, lessons explicitly indicate when fluency or culminating standards are addressed. Anchor Problems and Tasks are designed to address both conceptual foundations of the skills as well as procedural execution. Problem Set sections for relevant standards include problems and resources that engage students in procedural practice and fluency development, as well as independent demonstration of fluency. Skills aligned to fluency standards also appear in other units after they are introduced in order to provide opportunities for continued practice, development, and demonstration.”

Opportunities to develop procedural skill and fluency with teacher support and/or guidance occurs in the Anchor Tasks, at the beginning of each lesson, the Problem Sets, during a lesson, and Fluency Activities. Examples Include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 7, Fluency Activities, Number Talks, Multi-Digit Whole-Number Multiplication. The teacher guides students in a Number Talk, focusing on multiplying multi-digit whole numbers. The materials state, “Number talks are an opportunity to solve computational problems using mental strategies. They are typically a sequence of related computations that allow students to apply strategies from earlier computations to later ones. Number Talks can be completed as a whole class or in a small group with the teacher.” A Number Talks example states, “Multiplying multi-digit numbers using mental strategies” includes “Doubling and Halving.” One set of numbers to engage in a Number Talk includes, “1 × 72; 2 × 36; 4 × 18; 8 × 9.” This activity helps students develop 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm).

• In Unit 6, Multiplication and Division of Decimals, Lesson 4, Anchor Task, Problem 1 states, “Solve. Show or explain your work. a. 60 × 3 = ; 60 × 0.3 = ; 60 × 0.03 = __. b. What do you notice about #1? What do you wonder? c. Use what you noticed in #1 to solve 600 × 0.3 and 600 × 0.03.” Guiding Questions prompt teachers to ask, “How are the factors in these problems similar? How are they different? How are the products in these problems similar? How are they different? Why does 6 tens times 3 tenths result in a product whose unit is ones? Why does 6 tens times 3 hundredths result in a product whose unit is tenths? Why are 60 × 0.3 and 600 × 0.03 equivalent? What about 60 × 3 and 600 × 0.3?” These problems and guiding questions help students develop 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Anchor Task, Problem 2, students multiply a fraction by a fraction. The problem states, “Solve. Show or explain your work. a. $$\frac{3}{5}×\frac{1}{2}$$ b. $$\frac{1}{12}×\frac{8}{9}$$ c. $$\frac{5}{9}×\frac{3}{5}$$ d. $$\frac{15}{4}×\frac{8}{5}$$.” The Guiding Questions prompt teachers to ask, “How can we use what we learned in Anchor Task #1 to compute $$\frac{3}{5}×\frac{1}{2}$$? When I got to the point of computing $$\frac{1×8}{12×9}$$ in Part (b), I thought multiplying 12 by 9 was kind of a pain. Is there some way I can simplify this fraction before computing that product? (Follow a similar process to compute parts (c) and (d), seeing if it’s possible to simplify before computing the product.) Is it easier to simplify before or after performing the computation? Why?” This problem provides an opportunity for students to develop fluency of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction).

The instructional materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Problem Sets and Homework can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding, are designed for independent completion. Fluency Activities may be completed independently or with a partner. Examples include:

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Target Task, students use equivalent fractions as a strategy to add and subtract fractions. The task states, “Find three fractions that are equivalent to each of the following fractions. Use pictures and an equation to explain why the fractions are equivalent. 1. $$\frac{5}{8}$$ 2. $$\frac{7}{4}$$”. These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NF.1 (Use equivalent fractions as a strategy to add and subtract fractions).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Target Task, students “Solve each problem in two different ways. Show or explain your work. 1. $$\frac{2}{3}×\frac{5}{6}$$ 2. $$\frac{4}{9}×\frac{3}{8}$$.” These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction).

• In Unit 6, Multiplication and Division of Decimals, Lesson 6, Target Task, Problem 3 states, “Solve. Show or explain your work. a. 0.35 × 0.4;  b. 2.02 × 4.2.” These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used).

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Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials provide opportunities, within Problem Set and Homework problems for students to independently demonstrate multiple routine and non-routine applications throughout the grade level. Problem Set, or student practice problems, and Homework can be completed independently during a lesson. Target Task, or end of lesson checks for understanding, are designed for independent completion. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 9, Anchor Task, Problem 2, students solve routine application problems involving multiplying multi-digit numbers (5.NBT.5). The problem states, “On Friday, April 13, 2018, there were 749 outbound flights each carrying approximately 101 passengers. If there are usually 58,112 daily outbound passengers from Logan, how many more passengers flew out of Logan that Friday than usual?” Guiding Questions for teachers include, “Can you draw something to represent this problem? Is our answer reasonable? Why or why not? Why do you think this was a record-breaking day? What usually happens around that time of year?”

• In Unit 5, Multiplication and Division of Fractions, Lesson 6, Anchor Task, Problem 1, students engage in non-routine application problems as they solve real world problems involving multiplication of fractions and mixed numbers (5.NF.6), and apply and extend previous understanding of multiplication to multiply a fraction or a whole number by a fraction (5.NF.4). The problem states, “Some of the problems below can be solved by multiplying \frac{3}{5}×15, while others need a different operation. Select the ones that can be solved by multiplying these two numbers. For the remaining, tell what operation is appropriate. In all cases, solve the problem (if possible) and include appropriate units in the answer. a. There are 15 people at a party. \frac{3}{5} of them are boys. How many people at the party are boys? b. Wesley ran 15 miles on Monday and \frac{3}{5} mile on Tuesday. How many miles did Wesley run? c. If each person at a party eats \frac{3}{5} of a pound of roast beef and there are 15 people at the party, how many pounds of roast beef are needed? d. Nathaniel has 15 cups of soup split into 5 equal-sized portions. He’ll serve three of the portions for dinner tonight. What is the total amount of soup, in cups, that Nathaniel will serve tonight? e. 15 students in the fifth grade want to play soccer. \frac{3}{5} of the students in fifth grade want to play basketball. How many students want to play either soccer or basketball? f. Helena is carpeting a long corridor. It is \frac{3}{5} yards wide and 15 yards long. How much carpeting, in square yards, does Helena need? g. Lisbeth has \frac{3}{5} of a pound of chocolate that she wants to share evenly with 15 people. How much chocolate will everyone get? h. Tiffany has $15. She spends \frac{3}{5} of her money on a teddy bear. How much money does she have left?” Guiding questions for teachers include, “How can you connect what you know about multiplication with whole numbers to multiplication with fractions? What is the whole that is being referred to in the problem? Why does looking for keywords in a problem not always work as a strategy? (You could use Part (e) as an example here - ‘of'’ does not imply that these quantities should be multiplied together.)” • In Unit 6, Multiplication and Division of Decimals, Lesson 2, Anchor Task, Problem 1, students engage in non-routine problems that involve adding, subtracting, multiplying, and dividing decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (5.NBT.7). The problem states, “Mr. Wynn, the art teacher, is creating a mural on the side of the school building. The mural will be 2 meters tall and 6 meters long. After finishing the mural, he decided he wants to add a small section to the side of it, and the new section is also 2 meters tall but just 0.4 meters long. How many square meters is Mr. Wynn’s mural? What was the total length of the mural Mr. Wynn created?” Guiding questions for teachers include, “What model can you draw to represent the problem? How can you record that work with equations? Does finding the area of each separate piece of the mural and adding those areas together give the same result as finding the overall length and width of the space Mr. Wynn needed to cover and finding the overall area? Why?” • Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Task, students engage in routine problems that involve graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation (5.G.2). The problem states, “Jessica has$15 saved. She earns $6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs$45. How many hours does Jessica need to babysit in order to be able to buy it?”  Guiding Questions for teachers to ask include, “How can we represent this situation on a coordinate grid? How much money does Jessica have before she does any babysitting? Where do you see that on the graph? What is the coordinate pair? What does the coordinate pair (2, 27) mean in this context? How much money does Jessica have after babysitting for 4 hours? Where do you see that represented in the graph? How long will it take Jessica to earn $45 to buy the video game? Where do you see that represented in the graph?” Materials provide opportunities, within Problem Set and Homework, and Daily Word Problems for students to independently demonstrate multiple routine and non-routine applications throughout the grade level. Target Task, or end of lesson checks for understanding, are designed for independent completion. Examples include: • In Unit 1, Place Value with Decimals, Lesson 3, Additional Practice, Word Problem Practice, students independently solve routine problems when comparing zeros in place values by using additive to compare with difference unknown (5.NBT.2). “Santiago owns a small bread company. Saturday his company baked 3,860 loaves of bread. On Sunday his company baked 4,820 loaves of bread. How many more loaves did they bake on Sunday than on Saturday?” • In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 15, Problem Set, Problem 4, students independently solve routine addition problems with decimal numbers to the hundredths place (5.NBT.7). The problem states, “Van Cortlandt Park’s walking trail is 1.02 km longer than Marine Park’s. Central Park’s walking trail is 0.242 km longer than Van Cortlandt’s. Marine Park’s walking trail is 1.28 km. If a tourist walks all 3 trails in a day, how many kilometers would he or she have walked?” • In Unit 5 Multiplication and Division of Fractions, Lesson 20, Target Task, students independently solve non-routine word problems that involve the division of whole numbers and fractions (5.NF.7). Problem 2 states, “There are 7 math folders on a classroom shelf. This is \frac{1}{3} of the total number of math folders in the classroom. What is the total number of math folders in the classroom?” • In Unit 6, Multiplication and Division of Decimals, Lesson 24, Homework, Problem 6, students independently solve routine problems involving converting among different-sized standard measurement units within a given measurement system (5.MD.1). The problem states, “Sara and her dad visit Yo-Yo Yogurt again. This time, the scale says that Sara has 14 ounces of vanilla yogurt in her cup. Her father’s yogurt weighs half as much. How many pounds of frozen yogurt did they buy altogether on this visit? Express your answer as a mixed number.” ##### Indicator {{'2d' | indicatorName}} The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. All three aspects of rigor are present independently throughout Grade 5. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include: • In Unit 1, Place Value with Decimals, Lesson 13, Anchor Task, Problem 1, students develop conceptual understanding of rounding decimals as they use number lines. The problem states, “a. The number 8.263 lies between 8 and 9 on the number line. Label all the other tick marks between 8 and 9. Is 8.263 closer to 8 or 9 on the number line? b. Which tenth is 8.263 nearest to on the number line? Determine what values the two outermost spots on the number line below should be to help you determine which tenth 8.263 is closest to. Then plot 8.263 to prove your answer. c. Which hundredth is 8.263 nearest to on the number line? Determine what values the two outermost spots on the number line below should be to help you determine which hundredth 8.263 is closest to. Then plot 8.263 to prove your answer. ” (5.NBT.4: Use place value understanding to round decimals to any place.) • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 1, Fluency Tasks, Number Bowling, students develop procedural skill and fluency by playing a game with cards to write and evaluate numerical expressions. The problem states, “In this fluency activity, students use digits chosen at random to create expressions equivalent to as many digits, 1-10, as possible to knock down those bowling pins.” (5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.) • In Unit 7, Patterns and the Coordinate Plane, Lesson 10, Target Task, students engage in solving application problems as they interpret coordinate values of points in the context of the situation. The problem states, “The line graph below tracks the water level of Plainsview Creek, measured each Sunday for 8 weeks. Use the information in the graph to answer the questions that follow. a. About how many feet deep was the creek in Week 1? b. According to the graph, which week had the greatest change in water depth? c. It rained hard throughout the sixth week. During what other weeks might it have rained? Explain why you think so.“ (5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.) Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Examples include: • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 20, Anchor Task, Problem 4, develop conceptual understanding and procedural skill and fluency as they solve real world problems using multiplication and division. The problem states, “On Saturday, the owner of a department store gave away a$15 gift card to every 25th customer. A total of 8,879 customers came to the store on Saturday. What is the total value of the gift cards the owner gave away? How many additional customers would need to have come in for another gift card to be given away?” Guiding Questions include, “Can you draw something to represent this problem? What does the remainder mean in the context of this problem? How would you interpret it? How did you determine how many more customers would need to have come in for another gift card to be given away? Is your answer reasonable? Why or why not?” (5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm and 5.NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• In Unit 5, Multiplication and Division of Fractions, Lesson 6, Problem Set, Problem 4, students develop conceptual understanding and application as they solve real world problems involving multiplication of fractions. The problem states, “Mrs. Diaz makes 5 dozen cookies for her class. One-ninth of her 27 students are absent the day she brings the cookies. If she shares the cookies equally among the students who are present, how many cookies will each student get?” (5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.)

• In Unit 7, Patterns and the Coordinate Plane, Lesson 8, Target Task, students develop procedural skill and fluency alongside application as they solve problems involving ordered pairs. The task states, “Jillian plotted points Q and R on a coordinate grid, as shown below. Jillian wants to plot point S so that when points Q, R, and S are connected they form the vertices of a right triangle. Write an ordered pair that represents where Jillian should plot point S.” (5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.)

#### Criterion 2.2: Math Practices

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

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Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the Unit Summary and specific lessons (Criteria for Success, Tips for Teachers, or Anchor Task notes).

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 19, Problem Set, Problem 3, students “Assess the reasonableness of an answer by rounding and/or using the relationship between multiplication and division to check answers (MP.1).”  The problem states, “ a. At a school carnival there is an egg toss. There are 314 students in the school. Twelve eggs are in one carton. How many cartons are needed so that each student gets an egg to try the egg toss? b. The principal wants to give every student two tries at the egg toss. How will this decision affect the number of cartons he needs to buy?”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 10, Anchor Task, Problem 2, students “Assess the reasonableness and/or correctness of an answer based on number sense (MP.1).” The problem states, “Joe is baking cookies. He needs a total of 2 cups of sugar for the recipe. Joe bought a $$4\frac{1}{2}$$ cup bag of sugar and has used $$2\frac{3}{4}$$ cups already. Without solving the problem, does Joe have enough sugar? Explain your thinking.” Guiding Questions include, “How can we use what we did in Anchor Task #1 to help us? Does Joe have enough sugar? How did you figure that out without actually solving the problem?”

• In Unit 5, Multiplication and Division of Fractions, Lesson 20, Problem Set, Problem 2, students “Understand when a problem calls for the use of division and when other operations are called for in a problem involving a unit fraction and a whole number. (MP.1, MP.4)” The problem states, “Larry had $$\frac{1}{2}$$ of a submarine sandwich left over from a field trip. He decided to give it to his 6 friends to share after school. How much of the original sandwich did each person get? Show with numbers, words, and a picture or diagram.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 18, Anchor Task, Problem 4, students “Assess the reasonableness of a solution by rounding to estimate or by checking a solution using multiplication (MP.1).” The problem states, “Act 4 (the sequel): Bella has 6.3 kilograms of berries. She packs 0.35 kilogram of berries into each container. She then sells each container for $2.99. How much money will Bella earn if she sells all the containers? a. Write an expression to determine how much money Bella will earn if she sells all the containers. b. Find the amount of money Bella will earn if she sells all the containers.” Guiding Questions include, “What quantities and relationships do we know? What is the question asking you to find out? How did you find the answer to the question? Did anyone find the answer differently? Is your answer reasonable? How do you know?” MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include: • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 2, Target Task, Problem 3, students “Interpret expressions without evaluating them (MP.2).” The problem states, “Which of the following expressions represents a number that is 3 times larger than the sum of 8105 and 186?; a. (8105 + 186) ÷ 3; b. 3 × (8105 + 186); c. 8105 + 186 ÷ 3; d. 3 × 8105 + 186.” • In Unit 3, Shapes and Volume, Lesson 5, Target Task, Problem 1, students “Reason abstractly and quantitatively to see that the dimensions can be multiplied together in any order and the volume will remain the same (MP.2).” The problem states, “The right rectangular prism below has a length of 4 units, a width of 3 units, and a height of 6 units. Select the three equations that can be used to find the volume of the prism. a. 12 × 6 b. 4 × (3 × 6) c. 7 × 6 d. (4 + 3) x 6 e. 4 × 3 × 6.” • In Unit 5, Multiplication and Division of Fractions, Lesson 11, Anchor Task, Problem 3, students, “Write a story context to match a given expression involving the multiplication of two fractions (MP.2). Write an expression to match a story context involving the multiplication of two fractions (MP.2).” The problem states, “One section of a beach has a total of 180 people. Of these 180 people, $$\frac{4}{9}$$ are wearing a hat and $$\frac{2}{5}$$ of the people wearing hats are also wearing sunglasses. How many people in that section of beach are wearing both a hat and sunglasses? a. Write an expression to determine how many people in that section of the beach are wearing both a hat and sunglasses. b. Find the number of people in that section of the beach that are wearing both a hat and sunglasses. c. Find the number of people in that section of the beach that are wearing a hat but not wearing sunglasses.” Guiding Questions include, “What can you draw to represent the problem? What expression can you write to represent the situation? Is there more than one correct expression? How can you use your expression to find the number of people in that section of the beach that are wearing both a hat and sunglasses? How can you find the number of people in that section of the beach that are wearing a hat but no sunglasses?” • In Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Tasks, students “Contextualize and decontextualize a coordinate pair based on the situation.” The task states, “Jessica has$15 saved. She earns $6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs$45. How many hours does Jessica need to babysit in order to be able to buy it?” Guiding Questions include, “How can we represent this situation on a coordinate grid? How much money does Jessica have before she does any babysitting? Where do you see that on the graph? What is the coordinate pair? What does the coordinate pair (2, 27) mean in this context? How much money does Jessica have after babysitting for 4 hours? Where do you see that represented in the graph? How long will it take Jessica to earn $45 to buy the video game? Where do you see that represented in the graph?” ##### Indicator {{'2f' | indicatorName}} Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP3 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes) and students engage with the full intent of the MP through a variety of lesson problems and assessment items. Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include: • In Unit 2, Multiplication and Division of Whole Numbers, Lesson 4, Problem Set, Problem 2, students “Multiply multiples of powers of ten with multiples of powers of ten.” The problem states, “Ripley told his mom that multiplying whole numbers by multiples of 10 was easy because you just count zeros in the factors and put them in the product. He used these two examples to explain his strategy. 7,000 × 600 = 4,200,000; 800 × 700 = 560,000. Ripley’s mom said his strategy will not always work. Why not? Give an example.” • In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Anchor Tasks, Problem 1, students, “Determine whether two fractions are equivalent using an area model, a number line, or multiplication/division.” The problem states, “Ms. Kosowsky makes brownies in two pans of the same size. She cuts the pans in the following way: [Pan A is cut in halves; Pan B is cut in eighths.] Ms. Kosowsky gives one brownie from Pan A to Ms. Kohler and keeps four brownies from Pan B for herself. Ms. Kohler thinks this isn’t fair since she got one brownie and Ms. Kosowsky got four. Ms. Kosowsky thinks it’s fair. Who do you agree with, Ms. Kosowsky or Ms. Kohler? Why?” Guiding Questions include, “How much of each brownie pan did each teacher get? Do you agree with Ms. Kosowsky or Ms. Kohler? Equivalent fractions are fractions that represent the same portion of the whole and the wholes are equal-sized. Are these two fractions equivalent? How can we represent that with an equation? What do you notice about the numerators and denominators of the equivalent fractions? How can you use the area models to explain why this happens? How can you represent this using multiplication or division?” • In Unit 5, Multiplication and Division of Fractions, Lesson 17, Homework, Problem 5, students “Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.” The problem states, “Lisa claims that when multiplying any number between 0 and 10 by 100, the product is greater than 100. What is a possible number that can be multiplied by 100 to show that Lisa’s claim is not correct?” ##### Indicator {{'2g' | indicatorName}} Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (Criteria for Success, Tips for Teachers, or Anchor Tasks). MP4: Model with mathematics, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students are given many opportunities to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically. They model with math as they work with support of the teacher and independently throughout the units. Examples include: • In Unit 2, Multi-Digit Multiplication and Division of Whole Numbers, Lesson 20, Anchor Task, Problem 4, students “Solve multi-step word problems involving all four operations, including those that require the interpretation of the remainder (MP.1, MP.4).” The problem states, “On Saturday, the owner of a department store gave away a$15 gift card to every 25th customer. A total of 8,879 customers came to the store on Saturday. What is the total value of the gift cards the owner gave away? How many additional customers would need to have come in for another gift card to be given away?” Guiding Questions include, “Can you draw something to represent this problem? What does the remainder mean in the context of this problem? How would you interpret it? How did you determine how many more customers would need to have come in for another gift card to be given away? Is your answer reasonable? Why or why not?”

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Problem Set, Problem 10, students “Solve multi-step word problems involving measurement conversions (MP.4).” The problem states, “Tanya bought 12 water bottles. Of those bottles, 5 hold 300 milliliters each and 7 hold 1.5 liters each. How much water, in liters, does Tanya buy?”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 10, Anchor Tasks, Problem 1, students “Answer simple word problems regarding data represented in a coordinate graph (MP.4).” The problem states, “a. The following ordered pairs show the weight of a typical male greyhound, a breed of dog, during the first 28 months of his life. Graph the corresponding points, then connect the points in the order they are given to form a line graph. b. What do you notice? What do you wonder?” Guiding Questions include, “How did you decide what scale to use for the x-axis? The y-axis? How can you label the x-axis? The y-axis? What title can you give it to convey what is represented on the coordinate grid? What do you notice about the points you plotted? What do you wonder? What can we learn about the typical male greyhound’s weight by looking at the graph, especially compared with looking at the same information in the table?”

MP5: Use appropriate tools strategically, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities throughout the materials, with support of the teacher or during independent practice, to identify and use a variety of tools or strategies that support their understanding of grade level math. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 7, Anchor Tasks, Problem 1, students multiply two-digit by three-digit products using the method of their choice (i.e., standard algorithm, partial products, or area model). The problem states, “Find the products using any method. Then assess the reasonableness of your answer. a. 814 x 39, b. 715 x 53, c. 78 x 266.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Homework, Problem 6, students “Generate equivalent fractions using an area model, a number line, or multiplication/ division (MP.5).” The problem states, “Sammie took a really long bike ride through the mountains. She planned on taking breaks at equal points along the ride, as shown below. Sammie is $$\frac{3}{4}$$ of the way along her bike ride. a. Explain how you can use the number line to show $$\frac{3}{4}$$. b. Write a fraction that is equivalent to $$\frac{3}{4}$$.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 15, Homework, Problem 2, students “Decide which generalized method for computing products of mixed numbers will be most efficient for a particular problem and use it to compute the product (MP.5).” The problem states, “Circle the solution strategy that is more efficient in each of the problems above. Then, explain any similarities you notice between problems for which one strategy was more efficient.”

##### Indicator {{'2h' | indicatorName}}

Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP6 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes), and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

Students attend to precision in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 1, Criteria For Success states, “Understand that the order of operations is a convention that says operations should be performed from left to right in the following order (MP.6): a. Grouping symbol; b. Multiplication and division; C. Addition and subtraction.” Target Task, Problem 1 states, “What is the value of this expression? 100 − [5 × (3 + 4)].”

• In Unit 3, Shapes and Volume, Lesson 1, Criteria for Success, teachers are provided the following criteria for the lesson, “Use correct units and notation when recording the volume of a three-dimensional figure, including cubic units, cubic centimeters, and cm3 (MP.6).” Anchor Tasks, Problem 3, students find the volume of figures, making sure to accurately label the units of measure. The problem states, “Construct a figure that is 1 centimeter tall, 2 centimeters wide, and 2 centimeters long. What is its volume? Construct another figure with the same volume. Does the following figure have the same volume? Why or why not?” Guiding Questions include, “What is the volume of the figure you constructed? What might other figures with the same volume look like? (Present an example like the ones shown below.) Does this figure have a volume of 4 cubic centimeters? Why or why not? (Present a non-example with gaps.) Does this figure have a volume of 4 cubic centimeters? Why or why not? Does the figure shown in Part (c) have a volume of 4 cubic centimeters? Why or why not? Use the definition of volume to support your argument.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 13, Criteria For Success, teachers are provided the following criteria for the lesson, “Add like units with decimals using an area model or the standard algorithm, aligning place values correctly (which results in the decimal points being aligned) (MP.5, MP.6).” Target Task states, “Solve. a. 2.40 + 1.8 = __ b. 36.25 + 8.67 = __.”

• In Unit 7, Patterns and The Coordinate Plane, Lesson 4, Criteria for Success, “Plot points whose x- and/or y-coordinate is/are not a multiple of the scale of the corresponding axis (i.e., the point will not be located on intersecting gridlines) (MP.6).” Anchor Tasks, Problem 1, students, with support, label points on a coordinate plane whose locations are not intersecting points on the grid. The problem states, “Geraldo is plotting points on the following coordinate plane. Geraldo needs to plot the following four points: (0.8,12) (1.6,18) (1,30) (2,3). Is the graph large enough to fit his points? How do you know? What would be a point that he couldn't fit on the coordinate plane above? How do you know? How would the graph have to change in order for that point to fit on it?” Guiding Questions include, “Can Geraldo fit all of those points on the coordinate grid? How do you know? What point couldn’t he fit on the coordinate grid? How do you know? Let’s say you had no more space to extend the coordinate grid. How would you have to change the coordinate grid in order to be able to fit the point (18,6)? What about (45,0.6)? What about (80,4)?”

Students have frequent opportunities to attend to the specialized language of math in connection to grade-level content as they work with support of the teacher and independently throughout the units. The “Tips for Teachers” sections provide teachers with an understanding of grade-specific language and how to stress the specialized language during the lesson. Examples include:

• Each Unit Overview provides a link to a Fifth Grade Vocabulary Glossary. The glossary contains a chart with the columns “Word” and “Definition.” The glossary contains a chart with the columns “Word” and “Definition.” Under the Definition column, is the mathematical definition and an example. For example, for the word equation the definition reads, “A math statement that has an equal sign” and includes an example that reads, “10 × 5 = 50 is an equation.”

• In Unit 2, Multiplication/Division of Whole Numbers, Lesson 1, Anchor Task, Problem 1, Guiding Questions include, “An expression is a mathematical phrase that contains symbols for operations (like plus, minus, etc.), numbers, and/or letters that represent unknowns, as opposed to an equation, which is two expressions that are equal to one another. How did Felix evaluate, or solve, the expression? How can we record what he did as separate equations?”

• In Unit 3, Shapes and Volume, Lesson 1, Tips for Teachers states, “It is unclear whether students need to be familiar with the exponential notation for volume, namely units unit3, cm3, in3, etc. There are no released items from standardized tests or other reliable sources (e.g., Illustrative Mathematics) that use this notation. However, since students have seen exponents in the context of powers of ten in Unit 1, it is reasonable to assume that they can make sense of it here. If you choose to introduce the notation, you can eventually relate it to the idea of units being repeatedly multiplied together and therefore the notation representing that idea (e.g., a prism with measurements 2cm, 3cm, and 4cm has a volume of 2cm x 3cm x 4cm, or (2 × 3 × 4) (cm × cm × cm), or 24cm^3.”

• In Unit 7, Coordinate Plane, Lesson 1, Tips for Teachers state, “An ordered pair is a pair of two things written in a certain order. A coordinate pair is a pair of two coordinates written in a certain order, x then y. This distinction is not important for Grade 5 students and thus the terms are used interchangeably throughout the unit.”

##### Indicator {{'2i' | indicatorName}}

Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

MP7: Look for and make use of structure, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities throughout the units to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 2, Criteria for Success states, “Write numerical expressions based on verbal/written descriptions of calculations (e.g., write 2 × (8 + 7) to express the calculation ‘add 8 and 7, then multiply by 2’) (MP.7). Write descriptions of calculations based on numerical expressions (e.g., write ‘add 8 and 7, then multiply by 2’ to describe the expression 2 × (8 + 7)) (MP.7).” In Anchor Tasks, Problem 2 states, “For each problem below, write an expression that records the calculations described below, but do not evaluate. a. 3 times the sum of 26 and 4; b. The quotient of 15 and 3 subtracted from 60.” Guiding Questions include, “How can you represent Part (a) with a picture? How can that picture help you write a numerical expression? Is there more than one correct way to write an expression for Part (a)? How many can we come up with? Why can we write expressions with the addends of the addition expression in either order? Or with the factors of the multiplication problem in either order? Why are parentheses necessary around the 26 and 4 in all of the expressions we wrote for Part (a)? How can you represent Part (b) with a picture? How can that picture help you write a numerical expression? Can we write expressions with the values of the subtraction expression in either order? Or with the values of the division problem in either order? Why not? Do we need parentheses anywhere in the expression in Part (b)? Why or why not?”

• In Unit 3, Shapes and Volume, Lesson 5, Criteria for Success states, “Look for and make use of structure to find the volume of concrete rectangular prisms by finding the number of cubes in a layer by multiplying its length times width, then multiplying by the number of layers (MP.7).” In Target Tasks, Problem 1 states, “Use the figure below to answer the following questions. 1. How many layers are in the figure to the right? b. How many cubes are in each layer? c. What is the volume of the figure? d. Explain how you could find the volume of the figure using a different number of layers.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 11, Criteria For Success states, “Use the properties of addition and subtraction to make a multi-term computation easier (e.g., to add \frac{1}{3}+\frac{3}{5}+\frac{2}{3}, first add \frac{1}{3} and \frac{2}{3} to make 1, making the computation easier) (MP.7).” In Problem Set, Problem 3 states, “Erin jogged 2\frac{1}{4} miles on Monday. Wednesday, she jogged 3\frac{1}{3} miles, and on Friday, she jogged 2\frac{2}{3} miles. How far did Erin jog altogether?”

• In Unit 6, Multiplication and Division of Decimals, Lesson 19, Criteria For Success states, “Write descriptions of calculations involving decimals based on numerical expressions (e.g., write ‘add 8 and 7, then divide 3 tenths by that sum’ to describe the expression 0.3 ÷ (8 + 7)) (MP.7).” In Target Task, Problems 1 and 2 state, “1. Write an equivalent expression in numerical form. Then evaluate it. Twice as much as the difference between 7 tenths and 5 hundredths; 2. Write an equivalent expression in word form. Then evaluate it. 9 ÷ (0.16 + 0.2).”

MP8: Look for and express regularity in repeated reasoning, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities throughout the materials, with support of the teacher or during independent practice, to use repeated reasoning in order to make generalizations and build a deeper understanding of grade level math concepts. Examples include:

• In Unit 1, Place Value with Decimals, Lesson 1, Criteria for Success states, “Look for and express regularity in repeated calculations of multiplying 10 by itself some number of times, and explain patterns in the number of zeros in the product (MP.8).” Problem Set, Problem 2, students use repeated reasoning and multiples of 10 to solve problems looking for and applying knowledge of patterns. The problem states, “Solve.

a. 10 × 10 × 10 × 10 = ___

b. 10 × 10 × 10 = ___

c. 10 × 10 × 10 × 10 × 10 × 10 = ___

d. 10 × 10 × 10 × 10 × 10 = ___.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Criteria For Success states, “Deduce the generalized method for multiplying a fraction times a fraction (MP.8).” Target Task states, “Solve each problem in two different ways. Show or explain your work. a. \frac{2}{3}×\frac{5}{6} 2. \frac{4}{9}×\frac{3}{8}.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 7, Criteria for Success states, “Deduce the general pattern about the placement of the decimal point when multiplying decimals, namely that the number of decimal places in the product is the sum of decimal places in each factor (MP.8).” Homework, Problem 7 states, “Gerard solves 2.34 × 9.2 by computing 234 × 92, then moving the decimal point three places to the left. Why does Gerard’s method make sense?”

### Usability

The materials reviewed for Fishtank Plus Math Grade 5 partially meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports, meet expectations for Criterion 2, Assessment, and partially meet expectations for Criterion 3, Student Supports.

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, include standards correlation information that explains the role of the standards in the context of the overall series, provide explanations of the instructional approaches of the program and identification of the research-based strategies, and provide a comprehensive list of supplies needed to support instructional activities. The materials do not contain adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

##### Indicator {{'3a' | indicatorName}}

Materials provide 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.

The materials reviewed for Fishtank Plus Math Grade 5 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, Addition and Subtraction of Fractions/Decimals, 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, “When referred to fractions and decimals throughout Units 4-6, use unit language as opposed to ‘out of’ or ‘point’ language (e.g.,$$\frac{3}{4}$$ should be described as ‘3 fourths’ rather than ‘3 out of 4’ and 0.8 should be described as ‘8 tenths’ rather than ‘zero point eight’). To understand why this is important for fractions (which can be extrapolated to decimals), read the following blog post: Say What You Mean and Mean What You Say by William McCallum on Illustrative Mathematics. Read the following table that includes models used in this unit.” Additionally, the Unit Summary contains Essential Understandings. It states, “Quantities cannot be added or subtracted if they do not have like units. Just like one cannot add 4 pencils and 3 bananas to have 7 of anything of meaning (unless one changes the unit of both to ‘objects’), the same applies for the units of fractions (their denominators) and the units of decimals (their place values). This explains why one must find a common denominator to be able to add fractions with unlike denominators and why one must align corresponding places correctly (which in turn aligns the decimal points) when adding and subtracting decimals. ‘It is not necessary to find a least common denominator to calculate sums of fractions, and in fact the effort of finding a least common denominator is a distraction from understanding algorithms for adding fractions’ (NF Progressions, p. 11). Similar to computational estimates with whole numbers, some computational estimates with fractions can be better than others, depending on what numbers are chosen to use in place of the actual values. Computational estimates can be too high or too low, depending on how the numbers were originally estimated, what computation is being performed, and where in the number sentence it’s located. ‘Because of the uniformity of the structure of the base-ten system, students use the same place value understanding for adding and subtracting decimals that they used for adding and subtracting whole numbers’ (NBT Progression, p. 19).”

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 1, Place Value With Decimals, Lesson 8, Tips for Teachers include guidance to address common misconceptions with decimal patterns. The materials state, “A common misconception when multiplying decimals by 10 is that one just ‘adds a zero’ to the end of the number, which with decimals does not change the value of the number. If you notice this misconception come up, address it directly with the whole class, using various models and arguments to dispute it. The Target Task should give the teacher helpful feedback on whether this misconception persists by the end of the lesson.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 2, Anchor Tasks Problem 1 Notes provide teachers guidance about how to set students up to solve the problems. The materials state, “The contexts for the problems were intentionally chosen to encourage students to use one type of model—an area model for part (a) and a number line for part (b). Students, of course, can use them interchangeably. Since students relied more heavily on tape diagrams and number lines for fraction addition and subtraction in Grade 4, they may gravitate towards those, in which case you can decide whether to introduce the area model here or wait for Lesson 4 when they become more useful to find a common unit. An area model and number line for Parts (a) and (b), respectively, are shown below. If students struggle to see why the numerators are added and the denominators stay the same, relate it to the idea of the denominators being the units of the fractions. Just like 3 bananas and 4 bananas together are 7 bananas, 3 eighths and 4 eighths together are 7 eighths.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 9, Tips for Teachers provide context when students divide decimals by a single-digit whole number. The materials state, “Because students will encounter cases of decimal division for which they have not seen the equivalent fraction division, students cannot use fraction division to reason about where to place the decimal point in all cases of decimal division that students will encounter. Thus, it is not included in this lesson since students have not yet performed the corresponding fraction division related to today’s cases. This lesson is analogous to Lessons 1 and 2 but with decimal division. It has been consolidated into one lesson, assuming that students are more easily able to use reasoning about the placement of the decimal point. However, if students would benefit from this lesson being broken up into two days, including if they may benefit from it based on students’ reasoning skills when placing the decimal point in a decimal quotient or based on their procedural comfort with multi-digit division, it can be split into two lessons.”

##### Indicator {{'3b' | indicatorName}}

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials reviewed for Fishtank Plus Math Grade 5 partially 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.

While adult-level explanations of concepts beyond the grade are not present, Tips for Teachers, within some lessons, can support teachers to develop a deeper understanding of course concepts. Examples include:

• In Unit 1, Place Value with Decimals, Lesson 10, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “Comma placement in a number signals how to read that number. As Van de Walle says, “to read a number, first mark it off in triples from the right. The triples are then read, stopping at the end of each to name the unit (or cube shape) for that triple. Leading zeros in each triple are ignored when the number is read. If students can learn to read numbers like 059 (fifty-nine) or 009 (nine), they should be able to read any number” (Van de Walle, Student-Centered Mathematics, 3–5, p. 166). Note, of course, that this only applies to the whole number portion of a decimal number. “Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as 'zero point one five' or 'point one five.' (Decimals smaller than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this manner. (For example, the number $$\pi$$, which has infinitely many non-zero digits, begins 3.1415…) Other ways to read 0.15 aloud are '1 tenth and 5 hundredths' and '15 hundredths,' just as 1,500 is sometimes read '15 hundred' or '1 thousand, 5 hundred.' Similarly, 150 is read 'one hundred and fifty' or 'a hundred fifty' and understood as 15 tens, as 10 tens and 5 tens, and as 100 + 50. Just as 15 is understood as 15 ones and as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth and 5 hundredths in computations with decimals” (NBT Progression, p. 14). For the sake of avoiding confusion, we recommend only using the word “and” in place of the decimal point and nowhere else. For example, 217.35 is read “two hundred seventeen and thirty-five hundredths,” not “two hundred and seventeen and thirty-five hundredths.” Further, to build student understanding of decimal values, we recommend refraining from using the “point one five” language until you are sure students have a strong sense of place value with decimals.”

• In Unit 3, Shapes and Volume, Lesson 6, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “Lesson 6’s three-act task addresses the idea of “filling” volume. As the Geometric Measurement Progression states, “solid units are ‘packed,’ such as cubes in a three-dimensional array, whereas a liquid ‘fills’ three-dimensional space, taking the shape of the container… The unit structure for liquid measurement may be psychologically one-dimensional for some students” (GM Progression, p. 26). In addition to addressing the idea of filling volume and the use of the formula v = l w h to solve a contextual problem, Lesson 6’s three-act task also relies somewhat on the idea that volume is additive, which provides a nice preview of tomorrow’s work on that concept. For that reason, focus the lesson on the relationship between the length, width, and height and the volume. Or, you may decide to have students work on the Problem Set before the three-act task, which can provide a transition between thinking of volume in these two ways.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 20, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “There are two interpretations for division: (a) partitive division, also called equal group with group size unknown division, and (b) measurement division, also called equal group with number of groups unknown. In Grade 5, students apply and extend this understanding of the two types of division with whole numbers to divide unit fractions by whole numbers and whole numbers by unit fractions. To develop an understanding of the division of a unit fraction by a whole number, they use partitive division, such as in the problem “$$\frac{1}{2}$$ meter of cloth is cut into three equal pieces. How long is each piece of fabric?" Inversely, to develop an understanding of the division of a unit fraction by a whole number, they use measurement division, such as in the problem, “Three meters of cloth are cut into $$\frac{1}{2}$$ meter strips. How many strips are cut?” That way, as Bill McCallum notes, “students can build on their understanding of whole number division without having to grapple with fractional groups, so long as they understand both of these interpretations of division” (“Fraction Division Part 2: Two Interpretations of Division”, Mathematical Musings). Thus, students are exclusively given partitive division problems in Lesson 18 and measurement division problems in Lesson 19 to help them build a strong conceptual understanding of fraction division before seeing other examples of types of division problems in Lesson 20.”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 1, Tips for Teachers, contain adult-level explanations of complex grade level concepts. “Even when lessons do not call for it, students may benefit from using graph paper to aid in their precision (MP.6), though they may not need it. Make it available for students to use but without explicitly requiring the use of the tool, allowing them to use appropriate tools strategically (MP.5). An ordered pair is a pair of two things written in a certain order. A coordinate pair is a pair of two coordinates written in a certain order, x then y. This distinction is not important for Grade 5 students and thus the terms are used interchangeably throughout the unit.”

##### Indicator {{'3c' | indicatorName}}

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

The materials reviewed for Fishtank Plus Math Grade 5 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 5th 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 5th Grade Math, Unit 2, Multiplication and Division of Whole Numbers, 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 3, Shapes and Volume, Lesson 1, the Core Standard is identified as 5.MD.C.3 and 5.MD.C.4. The Foundational Standard is identified as 3.MD.C.5. 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 with Decimals, Unit Summary includes an overview of how the math of this unit builds from previous work in math. The materials state, “In Grade 4, students developed the understanding that a digit in any place represents ten times as much as it represents in the place to its right (4.NBT.1). With this deepened understanding of the place value system, students read and wrote multi-digit whole numbers in various forms, compared them, and rounded them (4.NBT.2—3).”

• In Unit 3, Shapes and Volume, Unit Summary includes an overview of the Math Practices that are connected to the content in the unit. The materials state, “Throughout Topic A, students have an opportunity to use appropriate tools strategically (MP.5) and make use of structure of three-dimensional figures (MP.7) to draw conclusions about how to find the volume of a figure.”

• In Unit 7, Patterns and the Coordinate Plane, Unit Summary includes an overview of how the content in 5th grade connects to mathematics students will learn in middle grades. The materials state, “This work is an important part of “the progression that leads toward middle-school algebra” (6—7.RP, 6—8.EE, 8.F) (K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics, p. 7). This then deeply informs students’ work in all high school courses. Thus, Grade 5 ends with additional cluster content, but that designation should not diminish its importance this year and for years to come.”

##### Indicator {{'3d' | indicatorName}}

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement. While curriculum resources support teachers with planning, instruction, and analysis of student progress, there are no specific resources for parents or caregivers.

##### Indicator {{'3e' | indicatorName}}

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Fishtank Plus Math Grade 5 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.”

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. 5th Grade, 5.NBT.5, 5.OA.1, 5.NBT.3, 5.NBT.4, 5.NBT.5, 5.NBT.6, 5.NBT.7, 5.NF.1, 5.NF.4, 5.NF.7, and 5.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

• 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.”

##### Indicator {{'3f' | indicatorName}}

Materials provide a comprehensive list of supplies needed to support instructional activities.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 5th Grade Course Summary, Course Material Overview, Course Material List 5th Grade Mathematics states, “The list below includes the materials used in the 5th 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:

• Tape or staplers are used in Units 1 and 3, one per group. In Unit 1, Place Value and Decimals, Lesson 1, students work with place value to millions. The materials state, “For this task, the teacher will need paper hundreds flats (20 count) cut to show 1 one, 1 ten, 1 hundred. Students will need a lot of copies of the paper hundreds flats (20 count), cut along the lines to make hundreds, and tape or staples to group the hundreds together.”

• Base-ten blocks are used in Unit 1, maximum of 80 ones, 70 tens, 80 hundreds, and 8 thousands per individual, pair, or group of students.

• Centimeter cubes are used in Unit 3, two hundred per pair or group of students.

• Cardstock is used in Unit 3, three sheets per student.

• A pair of scissors is used in Unit 5, one per student. In Unit 5, Multiplication and Division of Fractions, Lesson 1, students model fractions as division using area models. The materials state, “Students may need pieces of paper and scissors for this task (optional: see note below).”

• Markers or crayons are used in Units 3 and 7, three different colors per student.

• Graph paper is used Unit 7, two sheets per student.

##### Indicator {{'3g' | indicatorName}}

This is not an assessed indicator in Mathematics.

##### Indicator {{'3h' | indicatorName}}

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for Assessment. The materials include an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up, and the materials provide assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices. The materials include assessment information in the materials to indicate which standards and mathematical practices are assessed.

##### Indicator {{'3i' | indicatorName}}

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Fishtank Plus Math Grade 5 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 with Decimals, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 10 is aligned to 5.NBT.3b and states, “Select the two correct comparisons. A. 0.057 < 0.008, B. 0.057 < 0.57, C. 0.57 = 0.570, D. 0.57 > 1.001, E. 0.057 < 0.049.”

• In Unit 2, Multiplication and Division of Whole Numbers, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 5 is aligned to MP4 and states, “The ground floor of a hotel measures 74 feet long and 126 feet wide. There is 15 times as much square footage in the whole hotel as on the ground oor. What is the total square footage of the hotel? Show or explain your work.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Unit Summary, Unit Assessment, Answer Key denotes standards addressed for each question. Problem 6 is aligned to 5.NF.1 and states, “Solve. 7\frac{5}{6} + 3\frac{4}{9}. Select the two correct answers. A. 10\frac{15}{54}, B. 10\frac{9}{15}, C. 11\frac{15}{54}, D. 10\frac{5}{18}, E. 11\frac{5}{18}, F. 10\frac{3}{5}.”

• In Unit 6, Multiplication and Division of Decimals, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 1 is aligned to MP2 and states, “At a café, the cost of a turkey sandwich is 1 less than twice the cost of a side salad. A side salad costs 3.50. Which of the following expressions can be used to find the cost, in dollars, of a turkey sandwich at the café? A. $$3.50\times2−1$$, B. 3.50\times2+1, C. (3.50−1)\times2, D. (3.50+1)\times2.”

• In Unit 7, Patterns and the Coordinate Plane, Unit Assessment Answer Key includes a constructed response and 2-point rubric with the aligned grade-level standard. Problem 5d is aligned to 5.G.2 and states, “The data table below shows the height of a typical meerkat at different times during their first 20 months of life. Decide whether the meerkat grew more from month 0 to month 10 or from month 10 to month 20. Explain your answer.”

##### Indicator {{'3j' | indicatorName}}

Assessment system 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.

The materials reviewed for Fishtank Plus Math Grade 5 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, “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 6, Multiplication and Division of Decimals, Lesson 24, Target Task, Problem 2 states, “Solve. Show or explain your work. Yolanda took a bus to visit her grandmother. She brought a CD to listen to on the bus. The CD is 78 minutes long. The bus ride was 2\frac{1}{2} hours long. How many minutes longer was the bus ride than the CD?” A Mastery Response is provided, which states, “$$2\frac{1}{2}$$ x 1hr = $$2\frac{1}{2}$$ × 60 minutes =120 + 30 minutes = 150 minutes. 150 - 78 = 72. The bus ride was 72 minutes longer than the CD.”

##### Indicator {{'3k' | indicatorName}}

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for Fishtank Plus Math Grade 5 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 1, Place Value with Decimals, Post-Unit Assessment, Problem 6, supports the full development of MP3 (Construct viable arguments and critique the reasoning of others). The materials state, “Terry made an error while finding the product 0.4 × 100. He writes, When I multiply by ten I just add a zero to the end of my number, so since multiplying by 100 is the same as multiplying by ten twice, I add two zeros to the end of my number. So, 0.4 × 100 = 0.400. Identify Terry’s mistake. Explain what he should do to get the correct answer and include the correct answer in your response.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Post-Unit Assessment, Problems 2, 5, and 7 develop the full intent of 5.NBT.7 (Add, subtract, multiply and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction). Problem 2 states, “The decimal grids below are shaded to model and expression. What is the value of the expression modeled by the decimal grids? A. 3.29; B. 3.32; C. 4.10; D. 4.13.” Problem 5 states, “Fred is going to the movies. A movie ticket costs $15, and Fred wants to buy a bucket of popcorn for$3.50 and a candy bar for $0.89. Fred has$20. Does Fred have enough money for a movie ticket, a bucket of popcorn, and a candy bar? If so, how much change will he receive? If not, how much more money does he need? Show or explain your work.” Problem 7 states, “Solve. 14.4 - 5.63.” Additionally, in Unit 6, Multiplication and Division of Decimals, Mid-assessment, Problem 2 states, “Multiply or divide. Show or explain your work. a. 8 × 4.35 b. 3.89 ÷ 5.”

• In Unit 5, Multiplication and Division of Fractions, post assessment, Problem 12, supports the full development of MP2 (Reason abstractly and quantitatively, as students interpret multiplication as scaling). The materials state, “The students in Raul’s class were growing grass seedlings in different conditions for a science project. He noticed that Pablo’s seedlings were 1\frac{1}{2} times as tall as his own seedlings. He also saw that Celina’s seedlings were \frac{3}{4} as tall as his own. Which of the seedlings below must belong to which student? Explain your reasoning.”

• In Unit 7, Patterns and the Coordinate Plane, Mid-Unit Assessment, Problem 2 and Post-Unit Assessment, Problems 4 and 5, meet the full intent of 5.OA.3 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation). Mid-unit Assessment Problem 2 states, “Sonia and Ahmal plotted the point F on the coordinate grid below. a. Sonia wants to plot a point G so that F and G form a horizontal line. Write an ordered pair that represents where Sonia could plot point G. b. Ahmal wants to plot a point H so that F and H form a vertical line. Write an ordered pair that represents where Ahmal could plot point H.” Post-Unit Assessment, Problem 4 states, “Select the three statements that correctly describe the coordinate system. A. The x- and y-axes intersect at 10. B. The x- and y-axes intersect at the origin. C. The x- and y-axes are parallel number lines. D. The x- and y-axes are perpendicular number lines. E. The x- and y-coordinates are used to locate points on a coordinate plane.” Problem 5 states, “The data table below shows the height of a typical meerkat at different times during their first 20 months of life. A. Graph the data on the grid below. B. How many inches did the meerkat grow between month 4 and month 12? C. How many months did it take for the meerkat to grow from 7 inches to 12 inches?”

##### Indicator {{'3l' | indicatorName}}

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

According to Math Teacher Tools, Assessment Resource Collection, “Each post-unit assessment includes approximately 6-12 problems for Grades 3-5 and 10-16 problems for Grades 6-8. It is recommended that teachers administer the post-unit assessment soon, if not immediately, after completion of the unit. The assessment is likely to take a full class period.” While all students take the assessment, there are no recommendations for potential student accommodations.

Math Teacher Tools contain extensive information about strategies to utilize with sections, “Special Populations” and “Supporting English Learners.” One of many strategies includes, “Provide a prompt for students to respond to: Offering a scaffolded starting point for students to explain their thinking can be greatly beneficial to students who struggle in this area. This might look like providing sentence stems.”

Additionally, in Teacher Tools, Math, Special Populations, Strategies For Supporting Special Populations, Memory, Lesson Level Adjustments states, “Provide tools: Consider allowing the use of tools like multiplication charts and calculators when appropriate. This would be especially appropriate if the skill to be introduced that day is not directly about assessing students’ understanding of math facts/arithmetic but this skill is an underlying skill preventing them from being successful that day.” This type of guidance is absent from actual assessments

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Fishtank Plus Math Grade 5 partially meet expectations for Student Supports. The materials provide extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity and 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 partially provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics, and the materials partially provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

##### Indicator {{'3m' | indicatorName}}

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

he materials reviewed for Fishtank Plus Math Grade 5 partially meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics. There are general strategies and supports included for teachers, but regular and active participation of special populations is not enhanced with specific recommendations linked to daily learning objectives, standards, and/or tasks within grade-level lessons.

Within Math Teacher Tools, there is a Special Populations folder that includes resources to support teachers. According to the materials, “In this Teacher Tool, we aim to provide teachers with resources to 1) broaden their own understanding of learning disabilities related to areas of cognitive functioning, 2) reflect on how the content or demands of a unit or lesson may require modifications or accommodations, and 3) identify and incorporate specific strategies meant to support students with learning disabilities.” There are many suggestions for supporting special populations within three categories in the Math Teacher Tools, “Areas of Cognitive Functioning, Protocols for Planning for Special Populations, and Strategies for Supporting Special Populations.'' For example, in Strategies for Supporting Special Populations, Conceptual Processing, Lesson Level Adjustments states, “Use manipulatives: Incorporate opportunities to use manipulatives that illuminate mathematical concepts in addition to those already included in the curriculum. Some excellent options that can be applied to elementary and middle/high school include base ten blocks, two-color counters, unit squares and unit cubes (such as centimeter cubes), fraction strips/tiles, and algebra tiles. With this strategy, ensure your manipulatives highlight the key concept and eliminate all other distractions. When introducing manipulatives, be sure to model how to use the materials correctly, what each represents, etc.”

##### Indicator {{'3n' | indicatorName}}

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials reviewed for Fishtank Plus Math Grade 5 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, Multiplication and Division of Whole Numbers, Lesson 1, Problem Set, Problem 7 states, “CHALLENGE: Use the digits 0 to 9 to make the equation true. You don't need to use all the digits from 0 to 9, but you can only use each digit once. 20 = ___ + (___ - ___) × ___.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Anchor Tasks, Problem 4, Notes state, “This task is optional, depending on whether you have time for it. Students saw tasks like these in Grade 4 and they are included on the Problem Set, but students’ ability to solve tasks like this are not a necessary skill for fraction addition and subtraction (since they will always know the values they are multiplying the numerator and denominator by rather than need to figure them out). But, because they are more challenging than the fractions in Anchor Task #3 and include larger numbers, they encourage students to use the algorithm rather than draw models, which will help them fluently find equivalent fractions in the context of addition and subtraction of fractions with unlike denominators.”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 7, Problem Set, Problem 4e states, “CHALLENGE: i. Compare the coordinates of points Q and T. What is the difference of the x-coordinates? The y-coordinates? ii. Compare the coordinates of points Q and R. What is the difference of the x-coordinates? The y-coordinates? iii. What is the relationship of the differences you found in parts (e) and (f) to the triangles of which these two segments are a part?”

##### Indicator {{'3o' | indicatorName}}

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Fishtank Plus Math Grade 5 provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

Students engage with problem-solving in a variety of ways within the Anchor Problems, Problem Sets, and Target Tasks and Academic Discourse is a key component for the program. According to Math Teacher Tools, “Academic discourse is necessary for students to develop the critical thinking skills needed to construct viable arguments and critique the reasoning and ideas of others (Standard for Mathematical Practice 3). Academic discourse pushes students toward deeper understanding of concepts and ideas, encourages logical reasoning and thinking, and requires students to reflect on their own thinking and understanding. It is also vital for developing academic language, vocabulary, and oral language and communication skills.” Examples of varied approaches include:

• In Unit 1, Place Value with Decimals, Anchor Tasks, Problem 2, students round decimal numbers to the place required.The materials state, “Round each of the following values to the place values mentioned. Record your answer with the ‘≈’ symbol. 49.67 to the nearest whole and tenth 9.935 to the nearest tenth and hundredth 4.3816 to the nearest whole, tenth, hundredth, and thousandth”

• In Unit 3, Shapes and Volume, Lesson 13, Problem Set, Problem 3, students show their knowledge of the hierarchy of quadrilaterals and the properties of certain quadrilaterals. The materials state, “a. List the properties shared by all squares. b. List properties shared by squares and rhombuses. c. List properties shared by squares and rectangles.”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 5, Additional Activities, Word Problem Practice states, “Ana read for 4 hours this week. She read 3 times as much this week as she did last week. How long, in minutes, did Ana read for last week?”

Each unit contains a Self-Assessment for students to monitor their own progress and reflect on what they have learned throughout a unit. Each self-reflection builds metacognitive skills as “students assess their own understanding of the skill mentioned in each statement on a scale from 1 to 5. Then, based on those responses, they describe the areas in which they feel most confident, the least confident, and the tools and resources they can use to improve in those areas of least confidence.” For example:

• In Unit 7, Decimal Fractions, Unit Summary, Student Self-Assessment provides students with the “I Can” statements that relates to the Common Core State Standards and a response scale of 1-Not Yet, 2, 3-Sometimes, 4, 5-All the Time. The materials state, “I can evaluate expressions that involve parentheses, brackets, and/or braces. (5.OA.A.1) I can use parentheses, brackets, and/or braces in equations. (5.OA.A.1) I can translate words into expressions. (5.OA.A.2) I can explain how changing an expression will change its value without actually calculating it. (5.OA.A.2) I can fluently multiply multi-digit whole numbers. (5.NBT.B.5) I can divide 2-, 3-, and 4-digit whole numbers by 2-digit whole numbers using a variety of strategies. (5.NBT.B.6) I can check the answer to a division problem using multiplication and addition. (5.NBT.B.6) I can prove my division calculations are correct using equations, rectangular arrays, and/or area models. (5.NBT.B.6) Reflection: I feel most confident in my ability to: I feel least confident in my ability to: Things I can do to improve in areas where I feel less confident include:.”

##### Indicator {{'3p' | indicatorName}}

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Fishtank Plus Math Grade 5 provide some opportunities for teachers to use a variety of grouping strategies.

While suggested grouping strategies within lessons are not consistently present or specific to the needs of particular students, there is some general grouping guidance within Fluency Activities. The Procedural Skill and Fluency, Fluency Activities state, “The fluency activities are designed to be facilitated as a whole class, though suggestions for how to make each activity adaptable for centers, independent or partner work, and/or asynchronous practice are included in their descriptions.” Examples include:

• In Unit 3, Shapes and Volume, Lesson 1, Fluency Activities, Count Em Up states, “In this fluency activity, students count unit squares to find the area of a flat figure (3.MD.C.6) or unit cubes to find the volume of a solid figure (5.MD.C.4). This fluency activity should be completed as a whole class or in a small group with a teacher.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 3, Fluency Activities, More or Less states, “In this fluency activity, Students determine whether a given expression is more or less than a certain benchmark, such as one or one half, without evaluating it. This fluency activity can be played as a whole class or in small groups (though it can also be adapted for students to play independently or in partners by providing students with computations to estimate and corresponding multiplication expressions with rounded factors).”

• In Unit 6, Multiplication and Division of Decimals, Lesson 3, Fluency Activities, High or Low states, “In this fluency activity, students determine whether a given expression has a value that's higher or lower than expressions that can be used to estimate it. This activity can be done as a whole class or in a small group with the teacher, though it could be adapted for students to play independently or in partners by providing students with computations to estimate and corresponding addition and subtraction expressions with rounded addends or minuend/subtrahend.”

##### Indicator {{'3q' | indicatorName}}

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials reviewed for Fishtank Plus Math Grade 5 partially meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

While there are resources within Math Teacher Tools, Supporting English Learners, that provide teachers with strategies and supports to help English Learners meet grade-level mathematics, these strategies and supports are not consistently embedded within lessons. The materials state, “Our core content provides a solid foundation for prompting English language development, but English learners need additional scaffolds and supports in order to develop English proficiency as they build their content knowledge. In this resource we have outlined the process our teachers use to internalize units and lessons to support the needs of English learners, as well as three major strategies that can help support English learners in all classrooms (scaffolds, oral language protocols, and graphic organizers). We have also included suggestions for how to use these strategies to provide both light and heavy support to English learners. We believe the decision of which supports are needed is best made by teachers, who know their students English proficiency levels best. Since each state uses different scales of measurement to determine students’ level of language proficiency, teachers should refer to these scales to determine if a student needs light or heavy support. For example, at Match we use the WIDA ELD levels; students who are levels 3-6 most often benefit from light supports, while students who are levels 1-3 benefit from heavy support.” Regular and active participation of students who read, write, and/or speak in a language other than English is not consistently supported because specific recommendations are not connected to daily learning objectives, standards, and/or tasks within grade-level lessons. Examples of strategies from Math Teacher Tools include:

• In Teacher Tools, Supporting English Learners, Scaffolds for English Learners Overview states, “English learners should be interacting with the same complex tasks as the rest of the class. The job of the teacher is to ensure that the proper scaffolds are in place to make sure that English learners can access the complex tasks. Scaffolds should provide additional supports while maintaining the rigor of the core task, not simplify or modify the core task. Scaffolds should be determined by the student’s English Language level and the task. We recommend the following types of scaffolds; sensory, graphic, interactive, and noticing cognates  to help support English learners. For example, a sensory scaffold may be Videos, Films and Audio. For lighter EL support: Show a short clip of an idea or concept to preview background information necessary to access a task. (For example, prior to learning about probability simulations, watch examples of simulations in action.) For heavier EL support: Show a short clip of an idea or concept to pre-teach key vocabulary prior to teaching a lesson. Video could be English or students’ home language.”

• In Teacher Tools, Math, Supporting English Learners, Oral Language Protocols, “There are adjusting oral language protocols for both light English Learner support and heavy English Learner support. For the light English Learner support: Provide sentence frames for students to use. Include sentence frames that require students to use a variety of sentence structures. Provide lists of key academic vocabulary to use when discussing a particular topic. Introduce and preview vocabulary words using the 7-step lesson sequence. Include visuals and gestures with all vocabulary words. Assign specific group roles to ensure equitable participation (timekeeper, notetaker, facilitator, etc.). To provide heavy English Learner support: Provide sentence frames for students to use. Sentence frames may be a variety of sentence structures. Strategically group students with others who speak the same home language. Allow students to complete the assignment in either English or in their home language. Provide students with answers (either on the back of the task, or in another location in the room) to allow partners to check if their partner has the correct answer. Provide more think time to allow students to build an effective argument. For oral turn and talk questions, give students a written version of the question to reference.” There are suggested oral language protocols that include: Turn and Talk, Simultaneous Round Table, Rally Coach, Talking Chips, Numbered Heads Together, and Take a Stand.

• In Teacher Tools, Supporting English Learners, Planning for English Learners, Overview states, “Teachers need a deep understanding of the language and content demands and goals of a unit in order to create a strategic plan for how to support students, especially English learners, over the course of the unit. We encourage all teachers working with English learners to use the following process to prepare to teach each unit. We acknowledge that this work takes time, but we believe it is necessary in order to best meet the diverse needs of students. The steps for INTELLECTUALLY PREPARING A UNIT are Step One: Unpack the Unit, Step Two: Set a Vision for Mastery, Step Three: Plan for Assessment and Mastery, Step Four: Take Ownership.We believe that teacher intellectual preparation, specifically internalizing daily lesson plans, is a key component of student learning and growth. Teachers need to deeply know the content and create a plan for how to support students, especially English learners, to ensure mastery. Teachers know the needs of the students in their classroom better than anyone else, therefore, they should also make decisions about where to scaffold or include additional supports for English learners. We encourage all teachers working with English learners to use the following process to prepare to teach a lesson. Step One: Determine a Vision for Mastery and Step Two: Build the Lesson.”

##### Indicator {{'3r' | indicatorName}}

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Fishtank Plus Math Grade 5 provide a balance of images or information about people, representing various demographic and physical characteristics.

While images are not used within materials, there are names that could represent a variety of cultures and problems include reference to specific roles, instead of pronouns that reference a specific gender identity. Lessons also include a variety of problem contexts to interest students of various demographic and personal characteristics. Examples include:

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 14, Problem Set, Problem 3 states, “Mrs. Fan wrote ‘5 tenths minus 3 hundredths’ on the board. Michael said the answer is 2 tenths because 5 minus 3 is 2. Is he correct? Explain your reasoning.”

• In Unit 5, Multiplication and Division of Fractions, Pre-Unit Assessment states, “Ronaldo read 72 pages in his book on Saturday. He read 8 times as many pages as Darren. Write an equation that can be used to find p, the number of pages Darren read. Then solve the equation to find the total number of pages Darren read. Show your work.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Target Task, Problem 1 states, “Owen lives 1.2 kilometers from school. Lucia lives 0.86 kilometers from school. Ignacio lives 90 meters from school. If Owen, Lucia, and Ignacio all walk to and from school, how far, in kilometers, did they all walk in total?”

• Other names that could represent a variety of cultures are represented in the materials, i.e., Deshawn, John, Phillis, Mr. Manetta, Xena, Elihu, Gordon, and Addison.

##### Indicator {{'3s' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide guidance to encourage teachers to draw upon student home language to facilitate learning.

Although the Math Teacher Tools, Oral Language Protocols provide general guidance for supporting students’ native language, there are no specific suggestions for teachers to facilitate daily learning that builds on a student’s multilingualism as an asset. Oral Language Protocols suggests, “When picking a protocol for partner work or small group work, it is important to think through how English learners will be grouped and what role they will play in a particular group. Depending on the demands of the task and situation, students can be grouped with native and proficient English speakers, other ELs, or by home language. English learners should interact with a variety of different speakers in a variety of situations.” Teacher materials do not provide guidance on how to garner information that will aid in learning, including the family’s preferred language of communication, schooling experiences in other languages, literacy abilities in other languages, and previous exposure to academic everyday English.

##### Indicator {{'3t' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

While About Us, Approach, Culturally Relevant, provides a general overview of the cultural relevance within program components, materials do not embed guidance for teachers to amplify students’ diverse linguistic, cultural, and/or social backgrounds to facilitate learning. The materials state, “We are committed to developing curriculum that resonates with a diversity of students’ lived experiences. Our curriculum is reflective of diverse cultures, races and ethnicities and is designed to spark students’ interest and stimulate deep thinking. We are thoughtful and deliberate in selecting high-quality texts and materials that reflect the diversity of our country.” While some diversity in names or problem contexts are present within materials, specific guidance to connect the mathematical goals with students’ funds of knowledge in a way that makes learning relevant or motivating for students, is absent.

##### Indicator {{'3u' | indicatorName}}

Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide supports for different reading levels to ensure accessibility for students.

While the Math Teacher Tools, Special Populations, Supporting Special Populations, Language section notes some general recommendations for supporting language and scaffolding vocabulary, there is nothing specific about reading levels. Guidance includes, “Implement group reading strategies: Call on students throughout the class to read problems aloud, allowing students who might struggle in this area to listen and focus on comprehension. Proactively mark-up the text: To ensure students are spending time on the thinking and learning of the fundamental math concept of the day, consider pre-annotating the text provided to students or providing definitions for words within the text that might be a barrier for students comprehending the text.” Within the Anchor Tasks Notes or Tips for Teachers, there are some suggestions to scaffold vocabulary or concepts to support access to the mathematics, but these do not directly address different student reading levels. Examples include:

• In Unit 1, Place Value With Decimals, Lesson 10, Tips for Teachers state, “‘Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as “zero point one five” or “point one five.” (Decimals smaller than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this manner. (For example, the number \pi, which has infinitely many non-zero digits, begins 3.1415…) Other ways to read 0.15 aloud are “1 tenth and 5 hundredths” and “15 hundredths,” just as 1,500 is sometimes read “15 hundred” or “1 thousand, 5 hundred.” Similarly, 150 is read “one hundred and fifty” or “a hundred fifty” and understood as 15 tens, as 10 tens and 5 tens, and as 100 + 50. Just as 15 is understood as 15 ones and as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth and 5 hundredths in computations with decimals’ (NBT Progression, p. 14). For the sake of avoiding confusion, we recommend only using the word ‘and’ in place of the decimal point and nowhere else. For example, 217.35 is read ‘two hundred seventeen and thirty-five hundredths,’ not ‘two hundred and seventeen and thirty-five hundredths.’ Further, to build student understanding of decimal values, we recommend refraining from using the ‘point one five’ language until you are sure students have a strong sense of place value with decimals.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 20, Anchor Tasks, Problem 2, Notes state, "Parts (a) and (d) can be solved by dividing 3 ÷ 12. Note that Part (a) is a measurement division problem, or number of groups unknown division problem and Part (d) is a partitive division problem, or group size unknown division problem. They will explore this idea further in Anchor Task #2. Part (e) can be solved by dividing 12÷3, so it will be worthwhile to discuss the expression that can be used to solve Part (e) and having students solve, since they otherwise will not have practice solving a word problem involving division of a unit fraction by a whole number in the Anchor Tasks. Note that this task may be difficult for students, especially English Language Learners, since distinguishing between the concepts of dividing in half and dividing by half is very subtle and can be confusing. This confusion may arise elsewhere in the unit, so be mindful of when it occurs, and make sure they are able to make the distinction. It may be helpful to ask, ‘Would you expect the solution to be greater or less than the original amount?’ to help them understand whether it is a situation that involves multiplication or division.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 17, Anchor Tasks, Problem 1, Notes state, “This particular task helps illustrate Mathematical Practice Standard 1 (MP.1), Make sense of problems and persevere in solving them. Problem solving is based upon students engaging in tasks in which a solution pathway is not known in advance. As fifth graders approach these problems, they will analyze them to make sense of what each is asking and decide on the best solution pathways. Students will notice that even though the same numbers are used in problems (a), (b), (c), (d) [and (e)], the contexts are very different, and it might not be obvious when reading the problem that division is necessary in each context. As students look for similarities/differences between the problems, they observe that problems (b), (c), and (d) ask the question ‘how many groups?’ while problem (a) asks ‘how many in each group?’ These observations will determine their entry points into each problem. Students may also conclude that how the remainder is reported is dependent upon the particular context of each problem.”

##### Indicator {{'3v' | indicatorName}}

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials reviewed for Fishtank Plus Math Grade 5 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 with Decimals, Lesson 6, Anchor Tasks, Problem 1 uses a place chart to help students explain patterns in the number of zeros of the quotient when dividing a whole number by powers of 10. The materials state, “For this task, teachers and students may need base ten blocks (about 30 ones, 30 tens, 30 hundreds, and 3 thousands per group/teacher) and a Millions Place Value Chart.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 3, Anchor Tasks, Problem 1, students use two-sided counters as manipulatives to model multiplying fractions by a whole number.

• In Unit 7, Patterns and the Coordinate Plane, Lesson 1, Anchor Tasks, Problem 2 uses Inch Grid Paper to introduce and “construct a coordinate plane and identify the coordinates of given points.”

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Fishtank Plus Math Grade 5 integrate some technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards, and the materials have a visual design that supports students in engaging thoughtfully with the subject. The materials do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, and the materials do not provide teacher guidance for the use of embedded technology to support and enhance student learning.

##### Indicator {{'3w' | indicatorName}}

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Fishtank Plus Math Grade 5 integrate some technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

While technology integration is limited, teachers and students have access to external technology tools and virtual manipulatives, like GeoGebra, Desmos, or other resources, as appropriate. Examples include:

• In Unit 1, Place Value with Decimals, Lesson 12, Anchor Tasks, Problem 1, Notes, Desmos Get Close to Me - Rounding [clotheslines], students use a Desmos applet to round numbers as an alternative to the Anchor Task.

• In Unit 5, Multiplication and Division of Fractions, Lesson 5, Tips for Teachers, “Parts of a Whole," from the North Carolina Department of Public Instruction, students have opportunities to use the applet to multiply a fraction by a whole number.

##### Indicator {{'3x' | indicatorName}}

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Fishtank Plus Math Grade 5 do not include or reference digital  technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

##### Indicator {{'3y' | indicatorName}}

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Fishtank Plus Math Grade 5 provide a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within units and lessons that supports learning on the digital platform.

• Each lesson follows a common format with the following components: Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Tasks, Problem Set & Homework, Target Task, and Additional Practice. The layout for each lesson is user-friendly as each component is included in order from top to bottom on the page.

• The font size, amount of directions, and language in student materials is appropriate.

• The digital format is easy to navigate and engaging. There is ample space in the Problem Sets, Homework, and Assessments for students to capture calculations and write answers. Teachers can pre-select material from suggested sources and print for students, making it easier to navigate pages.

While the visual layout is appealing, there are spelling and/or grammatical errors within the materials.

##### Indicator {{'3z' | indicatorName}}

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Fishtank Plus Math Grade 5 do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

While teacher implementation guidance is included for Anchor Tasks, Notes, Problem Sets, and Homework, there is no embedded technology, so teacher guidance for it is not necessary.

## Report Overview

### Summary of Alignment & Usability for Fishtank Plus Math | Math

#### Math 3-5

The materials reviewed for Fishtank Plus Math Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials partially meet expectations for Usability: meet expectations for Teacher Supports (Criterion 1), meet expectations for Assessment (Criterion 2), and partially meet expectations for Student Supports (Criterion 3).

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 6-8

The materials reviewed for Fishtank Plus Math Grades 6-8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, Grades 6 and 7, meet expectations for Usability: meet expectations for Teacher Supports (Criterion 1), meet expectations for Assessment (Criterion 2), and partially meet expectations for Student Supports (Criterion 3). In Gateway 3, Grade 8 partially meets expectations for Usability: meets expectations for Teacher Supports (Criterion 1), meet expectations for Assessment (Criterion 2), and partially meet expectations for Student Supports (Criterion 3).

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

## Report for {{ report.grade.shortname }}

### Overall Summary

###### Alignment
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###### Usability
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