## Fishtank Math

##### v1
###### Usability
Our Review Process

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Fishtank Math instructional materials can be accessed at https://www.fishtanklearning.org.

This review was conducted May - December 2019 and reflects the materials available during review.

Title ISBN Edition Publisher Year
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### Overall Summary

The instructional materials reviewed for Match Fishtank, Grade 6 meet expectations for alignment to the CCSSM. ​The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor. The materials meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

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

### Focus & Coherence

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for not assessing topics before the grade level in which the topic should be introduced.

##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations for assessing grade-level content and, if applicable, content from earlier grades. The materials do not assess topics before the grade level in which the topic should be introduced. Unit Assessments were examined for this indicator, and all materials are available digitally and through downloadable PDFs.

Examples of assessment items aligned to grade-level standards include:

• Unit 1 Test, Question 3, “Slater used 6 black Legos and 18 green Legos to build a tower. What was the ratio of the number of black Legos to the number of Legos in the tower? Answer choices: a. 1:3,  b.1:4, c. 1:6, d. 1:9.” (6.RP.1)
• Unit 1 Test, Question 7, “Wyatt hiked 6 miles in 2 hours. At this same rate, what is the total number of miles Wyatt could hike in 9 hours?” (6.RP.3.b)
• Unit 2 Test, Question 1, “Roya paid $48 for 12 cartons of orange juice. What is the unit rate per carton of orange juice that Roya paid? Answer choices: a.$3, b. $4, c.$6, d. $12.” (6.RP.2) • Unit 5 Test, Question 4, “Which phrase is a description of 2m +7? Answer choices: a. more than 2 times m, b. 2 more than 7 times m, c. 2 times the sum of 7 and m, d. 7 times the sum of 2 and m.” (6.EE.2.a) • Unit 6 Test, Question 2, “A shelf has four books on it. The weight, in pounds, of each of the four books on the shelf is listed below. ‘2.5, 3.2, 2.7, 2.3 Which inequality represents the weight, w, of any book chosen from the shelf? Answer choices: a. w > 2.3, b. w < 2.4 , c. w > 3.2 , d. w < 3.3.” (6.EE.b) #### Criterion 1.2: Coherence Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade. The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote at approximately 75% of instructional time to the major work of the grade. ##### Indicator {{'1b' | indicatorName}} Instructional material spends the majority of class time on the major cluster of each grade. The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for spending a majority of instructional time on 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 chapters (units, modules, topics, etc.) devoted to major work of the grade (including assessments and supporting work connected to the major work) is six out of eight units, which is approximately 75%. • The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 88 out of 118, which is approximately 75%. • The number of days devoted to major work (including assessments and supporting work connected to the major work) is 112 out of 143, which is approximately 78%. A lesson-level analysis is most representative of the instructional materials because the units contain major work, supporting work, and assessments. As a result, approximately 75% of the instructional materials focus on major work of the grade. #### Criterion 1.3: Coherence Coherence: Each grade's instructional materials are coherent and consistent with the Standards. The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The instructional materials are also consistent with the progressions in the standards and foster coherence through connections at a single grade. ##### Indicator {{'1c' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The instructional materials reviewed for Match Fishtank Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Supporting standards/clusters are connected to the major standards/clusters of the grade, for example: • Unit 1, Understanding and Representing Ratios, Lesson 3, Anchor Problem 3, 6.NS.4 connects with 6.RP.1 by using common factors in order to find equivalent ratios. “Pam and her brother both open savings accounts. Each begins with a balance of 0 dollars. For every$2 that Pam saves in her account, her brother saves 5 in his account. a) Determine a ratio to describe the money in Pam’s account to the money in her brother’s account. b) Create two equivalent ratios that describe the amount of money in Pam’s account and the amount of money in her brother’s account.” • Unit 3, Multi-digit and Fraction Computation, Lesson 5, Target Task, supporting standard 6.G.1 connects with 6.NS.1 through solving a real-world problem involving area. “There are other ways to think about division of fractions. Try these two questions. They both use division, but why? And how do you know what to divide by what? 1. The water level in the reservoir has gone down 2 1/2 feet in the last month and a half. How fast is the water level going down per month? 2. Farmer Schmidt owns 3/4 of a square mile of land. Her field is a rectangle. One side is 2/3 of a mile. How long is the other side?” • Unit 5, Numerical and Algebraic Expressions, Lesson 4, Target Task connects supporting standard 6.G.2 to 6.EE.2 as students use the formulas to find the volume and surface area of rectangular prisms. “A cube has 6 sides, each with an area of $$s^2$$ square units. The surface area of a cube is the total of all 6 sides and is represented by the formula $$S = 6s^2$$. Find the surface area of a cube with the side lengths below. a. s = 3 inches b. s = 1.2 cm c. s= 2/3 feet.” • Unit 7, Geometry, Lesson 6, the Target Task fosters coherence between the clusters as students apply the standards from 6.G.A and 6.RP to determine the area of the trapezoid. “Find the deck area around the pool. The deck area is the white area in the diagram.” ##### Indicator {{'1d' | indicatorName}} The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades. Instructional materials for Match Fishtank Grade 6 meet expectations that the amount of content designated for one grade level is viable for one year. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications. As designed, the instructional materials can be completed in 143 days. Included in the 143 days are: • 118 lesson days • 17 review/flex days • 8 assessment days Each unit is comprised of 12 to 18 lessons that contain a mixture of Anchor Problems, Problem Set Guidance, a Target Task, and a Mastery Response. These components align to the number of minutes needed to complete each part as provided in the Pacing Guide. Based on the pacing guide, the suggested lesson time frame is 60 minutes: • 5 - 10 mins Warm up • 25 - 30 mins Anchor Problems • 15 - 20 mins Problem Set • 5 - 10 minutes Target Task ##### Indicator {{'1e' | indicatorName}} Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades. The instructional materials for Match Fishtank Grade 6 meet expectations for the materials being consistent with the progressions in the standards. The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Content from prior or future grades is clearly identified and related to grade-level work. Prior grade knowledge is explicitly related to grade-level concepts. The “future standards” align the work with future grade-level standards. Each lesson provides the teacher with current standards and foundational standards which are identified under the “Standards” tab. Through the Unit Overview, Tips for Teachers, and Unit Summary, teachers are provided explicit connections to prior and future knowledge for each standard. The Unit Plan Summary section includes a list of foundational standards from earlier grades that are connected to the content standards addressed in that unit, as well as a list of future standards that relate. For example: • Unit 1, Understanding and Representing Ratios, “In fourth and fifth grade, students learned the difference between multiplicative and additive comparisons and they interpreted multiplication as a way to scale. Students will access these prior concepts in this unit as they investigate patterns and structures in ratio tables and use multiplication to create equivalent ratios. The work students do in this unit connects directly to Unit 2: Rates & Percent and re-appears in Unit 6: Equations and Inequalities when students analyze and graph relationships between independent and dependent variables. Beyond sixth grade, students extend their understanding of ratios and rates to investigate proportional relationships in seventh grade. This sets the groundwork for the study of functions, linear equations, and systems of equations, which students will study in eighth grade and high school.” • Unit 2, Unit Rates and Percent addresses 6.RP.2, 6.RP.3, 6.RP.3.b, 6.RP.3.c, 6.RP.3.d, and 6.RP.4. Foundational standards are “Covered in previous units or grades that are important background for the unit: 4.MD.1, 4.MD.2, 5.MD.1, 5.NF.3, 5,NF.4.a, 5.NF.4.b, 5.NF.5, 5.NF.5.a, 5.NF.5.b, 5.NF.6, 4.NF.4.c, 4.NF.6, and 5.NBT.6 from previous grades, and 6.RP.1 from a previous unit.” • Unit 5, Numerical and Algebraic Expressions, “In elementary school, students used variables to represent unknown quantities, and they evaluated and described numerical expressions without exponents. They used the commutative property to enhance their understanding of multiplication and addition, and they used the distributive property when modeling partial areas. All of these concepts come together and support student understanding in this sixth-grade unit. Immediately following this unit, sixth graders will start a unit, Equations and Inequalities, where they will use algebra to model and solve real-world problems. They will also revisit percentages using new skills with expressions and equations to efficiently solve percent problems. In seventh and eighth grades, students continue to simplify and solve more complex expressions and equations using the same tools learned in this unit.” Foundational standards (5.OA 1&2, 4.OA.3, 4.NBT.5, & 5.MD.b) as well as future standards (7.EE.1, 7.EE.4). Additionally, 6.EE.5, 6.EE.7 and 6.EE.9 are considered future standards for this lesson as they are not identified until the next unit, Unit 6: Equations and Inequalities. Lessons include connections between grade-level work, standards from earlier grades, and future knowledge. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, Illustrative Mathematics, and EngageNY, Great Minds. For example: • Unit 1, Understanding and Representing Ratios, Lesson 1 objective, “define ratio and use ratio language to describe associations between two or more quantities.” This lesson supports 6.RP.1 and links back to Foundational Standards from grade 4, 4.OA.2 and 4.MD.1, as evident in Anchor Problem 3, “Abigail mixed 2 cups of white paint with 6 tablespoons (T) of blue paint.” Students write at least four ratio statements to describe the situation. • Unit 2, Unit Rates and Percent, Lesson 2 includes Foundational Standards: 6.RP.1 and Future Connections: 7.RP.1, 7.RP.2, 7.RP.3. Lesson objective: define rate and unit rate and find rates from situations involving ratios. (6.RP.2, 6.RP.3b) • Unit 5, Numerical and Algebraic Expressions, Lesson 6, teachers are directed to a Problem Set from Engage NY, which moves students from writing fractions to writing algebraic expressions as fractions. Students begin with writing 1 ÷ 2 without the division sign, then a ÷ 2, then proceed to the Problem Set: Problem 1. “Rewrite the expressions using the division symbol and as a fraction. Answer choices: a. Three divided by 4, b. The quotient of m and 11, c. 4 divided by the sum of h and 7, d. The quantity x minus 3 divided by y. Problem 2. Draw a model to show that x ÷ 3 is the same as x/3.” (6.EE.2) The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The Anchor Problem(s) help students make sense of the mathematics of the lesson as outlined in the Criteria for Success and Objective by providing them multiple opportunities to engage in the grade-level content in meaningful ways. The Problem Set Guidance provides students the opportunity to work with problems in a variety of formats to integrate and extend concepts and skills. The Target Task is aligned to the Objective and designed to cover key concepts from the lesson and identify any misconceptions students have. It serves also as an indicator of student understanding or mastery of the Objective. For example: • Unit 1,Understanding and Representing Ratios, Lesson 16, Target Task, students solve a part:whole ratio problem using a tape diagram (6.RP.3). For example, the Target Task states, “When Carla looked out at the school parking lot, she noticed that for every 2 minivans, there were 5 other types of vehicles. If there are 161 vehicles, how many of them are minivans?” • Unit 3, Multi-Digit and Fraction Computation, Lesson 2, Anchor Problem states, “Leonard made 1/4 of a gallon of lemonade and poured all of it into 3 glasses, divided equally. How much lemonade is in each glass? Write a division problem and draw a visual model.” (5.NF.7) • Unit 4, Rational Numbers, Lesson 6, the Target Task states, “Christina is trying to order the numbers -3 and -2 1/2 from least to greatest. She makes the claim below. Christina’s claim: “I know that -2 1/2 is less than -3. So, -2 1/2 must be less than -3.” Is Christina correct in her thinking? Explain why or why not. Use a number line to support your reasoning.” (6.NS.6.c) • Unit 6, Equations and Inequalities, Lesson 7, Problem Set Guidance links to Illustrative Math, Fruit Salad, “A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 280 pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?” (6.RP.a.3, 6.EE.b.7) Standard 6.SP.5.b: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement, is not addressed in any lesson, although it is listed in the Unit Overview for Unit 8: Statistics. Prior knowledge is explicitly identified and linked to grade-level work. For example: • Unit 3, Multi-Digit and Fraction Computation, Lesson 1, Tips for Teachers, reviews multiplication and division concepts learned in elementary grades as it introduces the sixth grade standard. “This lesson is approaching 6.NS.1. It reaches back to concepts students learned in earlier grades around multiplication and division in order for students to be able to extend on these concepts in following lessons in the unit.” • Unit 4, Rational Numbers, Lesson 1 objective states, “Extend the number line to include negative numbers. Define integers.” (6.NS.6, 6.NS.6c). This is connected to prior knowledge of 3rd grade: Understand a fraction as a number on the number line and represent fraction on a number line diagram and use this knowledge to extend the number line to integers. (3.NF.2) • Unit 6, Equations and Inequalities, Lesson Overview, teachers are reminded that this lesson brings together several concepts and skills students have worked on throughout the year. For example, some of those concepts are as follows: tables of equivalent values (6.RP.3), writing equations (6.EE.1), plotting points (6.NS.8), and determining values in ratio relationships (6.RP.3). Foundation skills (that link to several lessons) are also identified. For example, some of those skills are as follows: 5.OA.3, 6.RP.3, and 6.NS.6.c. This lesson contains several links to prior lessons such as Unit 3, Lesson 5, Target Task, “There are other ways to think about division of fractions. Try these two questions. They both use division, but why? And how do you know what to divide by what?” Question 1: “The water level in the reservoir has gone down 2 1/2 feet in the last month and a half. How fast is the water level going down per month?” Question 2: “Farmer Schmidt owns 34 of a square mile of land. Her field is a rectangle. One side is 2/3 of a mile. How long is the other side?” (6.NS.1) ##### Indicator {{'1f' | indicatorName}} Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. The instructional materials for Match Fishtank Grade 6 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the standards. The materials include learning objectives that are visibly shaped by CCSSM cluster headings and problems and activities that connect two or more clusters in a domain or two or more domains, when these connections are natural and important. The Units are divided into Lessons focused on domains. Grade 6 standards are clearly identified in the Pacing Guide, Standard Map Document, and a CCSSM Lesson Map found in the Unit Summary of each Unit. Additionally, each lesson identifies the objectives that address specific clusters. Instructional materials shaped by cluster headings include the following examples: • Unit 1, Understanding and Representing Ratios, Lesson 1, Objective, “Define ratio and use ratio language to describe associations between two or more quantities.” (6.RP.A) • Unit 2, Unit Rates and Percent, Lesson 2, Objective, “Find unit rates and use them to solve problems.” (6.RP.A) • Unit 3, Multi-digit and Fraction Computation, Lesson 5, Objective, “Solve and write story problems involving division with fractions.” (6.NS.A) • Unit 5, Numerical and Algebraic Expressions, Lesson 3, Objective, “Use variables to write algebraic expressions.” (6.EE.A) • Unit 6, Equations and Inequalities, Lesson 8, Objective, “Define and identify solutions to inequalities.” (6.EE.B) • Unit 6, Equations and Inequalities, Lesson 12, Objective, “Write equations for and graph ratio situations. Define independent and dependent variables,” (6.EE.C) • Unit 7, Geometry, Lesson 10, Objective, “Find volume of rectangular prisms with whole number and fractional edge lengths using unit and fractional unit cubes.” (6.G.A) Instructional materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where the connections are natural and important. For example: • Unit 2, Ratios and Proportional Relationships, Lesson 2, 6.RP.2 and 6.RP.3.b are connected when students demonstrate an understanding of a unit rate through drawing a tape diagram. Anchor Problem 2 states, “Adam’s Fruit Farm also has 100 acres, but he grows more than just apples. Oranges take up 60 acres of his farm. What percent of Adam’s farm is oranges? What percent is not oranges? Draw a 10 x 1 tape diagram to model the situation.” • Unit 5, Numerical and Algebraic Expressions, Lesson 4, connects 6.EE.A and 6.NS.A as students evaluate expressions by multiplication and division of fractions and decimals. Anchor Problem 1 states, “A square prism is shown below. The formula $$V = s^2h$$ can be used to find the volume of the square prism. What is the volume of the prism when the side length of the base measures 1.5 inches and the height measures 8 inches?” • Unit 6, Equations and Inequalities, Lesson 12, connects 6.RP.A and 6.EE.C as students write equations in situations involving ratios. Anchor Problem 1 states, “A recipe for sugar cookies calls for 1 cup of sugar for every 2 cups of flour.” Questions posed: a. “Use the ratio to complete the table.” b. “If you know the number of cups of sugar, s, in the recipe, how can you determine the number of cups of flour to use? What equation represents this relationship?” c. “If you know the number of cups of flour, f, in the recipe, how can you determine the number of cups of sugar to use? What equation represents this relationship?” d. “In each equation, what is the independent variable and what is the dependent variable?” • Unit 7, Geometry, Lesson 9, 6.G.1 supports 6.NS.3 as students use their understanding of integers to represent polygons on the coordinate plane. Anchor Problem 1 states, “A new park is being built in the city. In the park, there will be a cemented walkway that will wind through the park. The walkway will be completely enclosed by a short, gated fence that will line the path on all sides of the path. Each square unit in the coordinate grid represents 1 square yard.” Questions posed: “a. The city budget includes enough funds to include 50 square feet of cement and 60 yards of fencing. Will the budget cover the necessary expenses for cement and fencing? Defend your answer. b. Two statues will be placed at point (-1,2) and (-1,-4). How far apart, in units, are the two statues?” • Unit 8, Statistics, Lesson 9 connect 6.SP.A and 6.SP.B as students begin to understand spread and variability of data sets. Anchor Problem 2 states, “Jamie is planning to cover a wall with red wallpaper. The dimensions of the wall are shown below. Questions posed: a. “How many square feet of wallpaper are required to cover the wall?” b. “Wallpaper comes in long rectangular strips that are 24 inches wide. If Jamie lays the strips of wallpaper vertically, how many strips will she use and how long will each strip be? Explain.” c. “If Jamie lays the strips of wallpaper horizontally, can she cover the wall without wasting any wallpaper? Explain.” ###### Overview of Gateway 2 ### Rigor & Mathematical Practices The instructional materials for Match Fishtank Grade 6 meet the expectations for rigor and the Mathematical Practices. The materials meet the expectations for rigor that students develop and demonstrate conceptual understanding, procedural skill and fluency, and application. The materials meet the expectations for Mathematical Practices, and attend to the specialized language of mathematics. ##### Gateway 2 Meets Expectations #### Criterion 2.1: Rigor Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application. The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop and independently demonstrate conceptual understanding, procedural skill and fluency, and application with a balance in all three. ##### Indicator {{'2a' | indicatorName}} Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The instructional materials for Match Fishtank Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. All units begin with a Unit Summary and indicate where conceptual understanding is emphasized, if appropriate. Lessons begin with Anchor Problem(s) that include Guiding Questions designed to help teachers build their students’ conceptual understanding. The instructional materials include problems and questions that develop conceptual understanding throughout the grade level, especially where called for in the standards (6.RP.A, and 6.EE.3). For example: • Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1, students develop conceptual understanding when introduced to ratios through the use of diagrams. An example of this is as follows: “In a recipe for oatmeal raisin cookies, the ratio of teaspoons of cinnamon to cups of raisins is 4:8. Draw a diagram to represent the quantities, and write two other ratio statements for the situation.” (6.RP.A) • Unit 1, Understanding and Representing Ratios, Lesson 4, introduces students to the concept of equivalent ratios. An example of this is Anchor Problem 1, “Are Heather and Audrey’s ratios equivalent? Explain how you know.” (6.RP.1) • Unit 2, Unit Rates and Percent, Lesson 4, Anchor Problem 1, students are given two different prices for jugs of honey. Anchor Problem 1 states, “Would you rather buy one 5-pound jug of honey for15.35, or three 1.5-pound bottles of honey for $14.39? Justify your answer.” (6.RP.3) • Unit 5, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 1 uses an area model to show the distributive property conceptually. “Two rectangles were combined to create a larger rectangle, as shown below. “Write as many expressions as you can to represent the area of the larger, outer rectangle.” Guiding question: “How do your expressions connect back to the area model?” (6.EE.3) • Unit 5, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 2 uses tape diagrams to discover the concept of using the distributive property to produce equivalent expressions, “The tape diagram represents the expression 3x + 4y. Draw a tape diagram that shows twice the value of 3x + 4y.” Guiding Questions, “What does it mean to take “twice the value” of an expression? What does this look like in a diagram? Rearrange your diagram to group together the same values. What property is this? How does grouping your diagram in this way help you write a new expression?” (6.EE.3) • Unit 6, Equations and Inequalities, Lesson 13, Anchor Problem, Question 1 states that students relate variables to the coordinate plane. Students use tables to discover relationships between dependent and independent variables and graph them appropriately: “Determine which variable is dependent and which variable is independent. Make a table showing the number of pencils for 3 – 7 packages. Plot the points in the coordinate plane. If Sarah has 168 pencils, how many packages did she purchase?” (6.EE.9) Grade 6 materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, Illustrative Mathematics, EngageNY, Great Minds, and others. For example: • Unit 1, Understanding and Representing Ratios, Lesson 2, Target Task, students are asked to draw a picture and name two ratios for each given situation: “To make papier-mâché paste, mix 2 parts of water with 1 part of flour. A farm is selling 3 pounds of peaches for$5. A person walks 6 miles in 2 hours.” (6.RP.A)
• Unit 3, Multi-digit and Fraction Computation, Lesson 2, Target Task, students use diagrams to develop the concept of division of fractions by whole numbers: “How much lemonade is in each glass? Write a division problem and draw a visual model” (6.NS.1)
• Unit 4, Rational Numbers, Lesson 3, Problems 1 & 2, (Open Up Resources: Grade 6, Unit 7, Lesson 1 ) give students an opportunity to work independently to demonstrate conceptual understanding of rational numbers by answering questions about temperature, elevation, and sea levels and in some cases to represent points on a vertical number line. “Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level. Drag the points marking the mountains and trenches to the vertical number line and answer the questions: a. Which point in the ocean is the lowest in the world? What is its elevation? b. Which mountain is the highest in the world? What is its elevation? c. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0? d. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.” (6.NS.5)
• Unit 5, Numerical and Algebraic Expressions, Lesson 7, Problem Set Guidance, (Open Middle, Equivalent Expressions 1): Students work independently to explore the use of whole numbers to create equivalent expressions: “Using the whole numbers from 1-9 in the boxes below, create two expressions that are equivalent to one another. You can use each whole number at most once.”  (6.EE.3)
• Unit 5, Numerical and Algebraic Expressions, Lesson 9, Target Task, students complete the following: “For each problem, draw a diagram to represent the expression. Then use the diagram to write an equivalent expression.  a. 4(2m + n)  b. 5x + 15.” (6.EE.3)
• Unit 6, Equations and Inequalities, Lesson 8, Target Task, students define and identify solutions to inequalities. Students are given a list of values and asked, “Which of the following values are solutions to the inequality 5x - 8 $$\lesseq$$ 42. Select all that apply.” (6.EE.5)
##### Indicator {{'2b' | indicatorName}}
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials for Match Fishtank Grade 6 meet the expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop fluency and procedural skills, for example:

• Every Unit begins with a Unit Summary, where procedural skills for the content is addressed.
• In each lesson, the Anchor problem(s) provides students with a variety of problem types to practice procedural skills.
• Problem Set Guidance provides students with a variety of resources or problem types to practice procedural skills.
• There is a Guide to Procedural Skills and Fluency under teachers tools and mathematics guides.

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level, especially where called for by the standards (6.NS.2, 6.NS.3, 6.EE.A).

For example, students independently demonstrate fluency:

• Unit 3, Multi-Digit and Fraction Computation, Lesson 9, Target Task, Problem 1, students practice fluently adding, subtracting and multiplying decimals. “Calculate the product: 78.93 × 32.4.” (6.NS.3)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 10, Target Task, students are given the opportunity to independently demonstrate procedural skills in division of multi-digit numbers using the standard algorithm by responding to the question, “Use the standard algorithm to solve 392,196 ÷ 87. Check your answer using multiplication.” (6.NS.2)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 11, Target Task, Problem 1, students use the division algorithm to develop and maintain fluency in dividing whole numbers and decimals. “Find the decimal value of 3 ÷ 50 using any strategy. Then find the quotient using long division and show the answers are the same." (6.NS.2 and 6.NS.3)
• Unit 5, Numerical and Algebraic Expressions, Lesson 2, Mathematics Exponent Experimentation 2 activity is recommended as a Problem Set for the objective to evaluate numerical expressions involving whole-number exponents. This task supports fluency as students practice working with operations, decomposing numbers, and recognizing perfect squares and perfect cubes: “Here are some different ways to write the number 16: a) $$2^4$$ b) $$12 - (2^1+2^2)+ 500/50$$ c) $$2^3 + 2^3$$ d) $$2/3 \times 48^1 - (1+3)^2$$ . Find at least three different ways to write each value below. Include at least one exponent in all of the expressions you write. a. 81 b. $$2^5$$ c. 64/9” (6.EE.1)
• Unit 5, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 2 uses students’ understanding of area models to compare the values of expressions using exponents and area models of squares and rectangles, for example: “Four expressions are shown below along with four area diagrams. Match each expression to a diagram. Then evaluate the expression and find the area of the diagram to demonstrate they are equivalent.”  (6.EE.1)

The instructional materials provide opportunities for students to independently demonstrate procedural skills. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, and EngageNY, Great Minds. For example:

• Unit 5, Numerical and Algebraic Expressions, Lesson 9, students generate equivalent expressions. For example, Problem Set Guidance, Open Middle Distributive Property, states, “Fill in the boxes below using the whole numbers 0 through 9 no more than one time each so that you can make a true equation.” (6.EE.4)
• Unit 6, Equations and Inequalities, Lesson 10. Students work toward developing procedural skills in writing inequalities for real-world conditions. Anchor Problem 3 states, “Two similar situations are described below. Situation A: A backpack can hold at most 8 books. Situation B: A backpack can hold at most 8 pounds. Draw a graph for each situation to represent the solution set. Compare and contrast the two graphs.” Problem Set Guidance provides additional practice. “Write an equation to represent each situation and then solve the equation. Andre drinks 15 ounces of water, which is 3/5 of a bottle. How much does the bottle hold? Use x for the number of ounces of water the bottle holds.” (6.EE.8)
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Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The instructional materials for Match Fishtank Grade 6 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications can be found in single and multi-step problems, as well as routine and non-routine problems.

In the Problem Set Guidance and Target Tasks, students engage with problems that have real-world contexts and opportunities for application, especially where called for by the standards (6.RP.3, 6.NS.1, 6.EE.7, 6.EE.9). The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge. Students have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, MARS Formative Assessment Lessons, Robert Kaplinsky, Yummy Math, EngageNY - Great Minds, and others.

Examples of routine application include, but are not limited to those that are familiar situations and/or are presented in the CCSSM Table1: Common Addition and Subtraction Situations and Table 2: Common Multiplication and Division Situations. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 3, “To make green-colored water, Brian mixes drops of green food dye and cups of water in a ratio of 4:3.  a. Draw a double number line to represent the ratio of drops of green food dye to cups of water. b. Use your double number line to find 2 equivalent ratios. c. Brian’s friend, Evan, uses a ratio of 20 drops of green food dye to 15 cups of water. Will Evan’s water be the same color green as Brian’s? Explain your reasoning.” (6.RP.3)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 6, Target Task, “You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is  1 1/2 miles away. You are timing your progress and find that you can travel 2/3 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit. Solve the problem with a diagram and explain your answer. Then write and solve an equation and show that it is the same as what you got in your diagram.?” (6.NS.1)
• Unit 6,  Equations and Inequalities, Lesson 3, Anchor Problem 2, “At a market, a farmer sells apples for $1.33 per pound. At the end of a weekend, the farmer made$74.48 from selling apples. Which equation can be used to determine x, the number of pounds of apples the farmer sold over the weekend.” (6.EE.7)
• Unit 6, Equation and Inequalities, Lesson 7, Anchor Problem 3, “The school librarian, Mr. Marker, knows the library has 1,400 books, but he wants to reorganize how the books are displayed on the shelves. Mr. Marker needs to know how many fiction, nonfiction, and resource books are in the library. He knows that there are four times as many fiction books as resource books. There are half as many nonfiction books as fiction books. a. If these are the only types of books in the library, how many of each type of book are in the library? b. Draw a tape diagram to represent the books in the library, and then write and solve an equation to determine how many of each type of book there are in the library.”  (6.EE.6, 6.EE.7)
• Unit 6, Equations and Inequalities, Lesson 13, Target Task, “Arian wants to save 20% of his paychecks in a savings account. a. Write an equation to represent the amount Arian should save, s from a paycheck in the amount of p dollars.  b. Create a table of values with at least three or four different paycheck amounts.  c. Plot the values in the coordinate plane to show the relationship between the amount Arian saves and the amount Arian earns. d. Explain how you could use your graph to find how much of a $60 paycheck Arian should put into his savings account.” (6.EE.9, 6.RP.3.a) Examples of non routine application include, but are not limited to real-world context that are unfamiliar, novel, and/or unrehearsed. For example: • Unit 2, Unit Rates and Percent, Lesson 14, Problem Set Guidance allows students to apply strategies, organize information and their workspace to keep track of their solution pathway. “Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?” One possible solution to this non-routine problem uses an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster and 6.EE.B. (6.RP.2, 6.RP.3, 6.RP.3.c, 6.RP.3.d) • Unit 3, Multi-Digit and Fraction Computation, Lesson 5, Anchor Problem 2, Handout 2, students solve and write story problems involving division with fractions. For example, a. “Make up and solve two of your own slicing problems. In Problem A, you should not have any cheese left over, and in Problem B, you must have some cheese left over. b. For each problem, you need to determine how much cheese you start off with: how long is your block of cheese? You also need to say how thick you want the slices of cheese to be—or you can decide how many slices you will need in total. Keep in mind that the thickness of each slice should be between 1/32 and 1/2 inches thick. c. After you create your problems, make a poster showing each problem and its solution. Each solution should include an explanation, at least one calculation, and a diagram.” (6.NS.1) • Unit 3, Multi-Digit and Fraction Computation, Lesson 6, Anchor Problem 1, students solve problems involving division with fractions. For example, “It requires 3/4 of a credit to play a video game for one minute. Emma has 7/8 credits. Can she play for more or less than one minute? Explain how you know. How long can Emma play the video game with her 7/8 credits? How many different ways can you show the solution?” (6.NS.1) The instructional materials in Grade 6 provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example: • Unit 1, Understanding and Representing Ratios, Lesson 18, Target Task, “In a bag of jelly beans, there are purple jelly beans (grape) and red jelly beans (cherry). For every 4 purple jelly beans, there are 7 red jelly beans. There are 902 jelly beans in the bag. How many of each flavor are there? Choose a strategy to solve the problem. Explain why you chose this strategy and how it shows your solution." (6.RP.3) • Unit 3, Multi-Digit and Fraction Computation, Lesson 3, Target Task, “A jar has 5 tablespoons of honey in it. One serving of honey is 3/4 of a tablespoon. How many servings of honey are in the jar?” (6.NS.1) • Unit 6, Equations and Inequalities, Lesson 1, Target Task, “Draw a tape diagram or balance for each equation or situation below. a. You purchase 4 gift cards, each in the same amount. You spend a total of$60. b. x + 6 = 18. c. 6x = 18.” (6.EE.6, 6.EE.7)
• Unit 6, Equations and Inequalities, Lesson 5, Target Task, Problem 2, “Lee filled several jars with 1/4 cup of water in each jar. He used a total of 8 cups of water. Let j represent the number of jars that Lee filled. Write and solve an equation to find out how many jars Lee filled.” (6.EE.7)
• Unit 6, Equations and Inequalities, Lesson 7, Target Task, “A town's total allocation for firefighters’ wages and beneﬁts in a new budget is $600,000. If wages are calculated at$40,000 per firefighter and beneﬁts at $20,000 per firefighter, write an equation whose solution is the number of firefighters the town can employ if they spend their whole budget. Solve the equation.” (6.EE.6, 6.EE.7) ##### Indicator {{'2d' | indicatorName}} Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade. The instructional materials for Match Fishtank Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Many of the lessons incorporate two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding. There are instances where all three aspects of rigor are present independently throughout the instructional materials. For example: • Unit 3, Multi-Digit and Fraction Computation, Lesson 2, Anchor Problem 1, students develop conceptual understanding for dividing fractions. “Leonard made 1/4 of a gallon of lemonade and poured all of it into 3 glasses, divided equally. How much lemonade is in each glass? Write a division problem and draw a visual model.” (6.NS.1) • Unit 4, Rational Numbers, Lesson 8, Anchor Problem 3, students write inequalities to compare rational numbers in real-world contexts to develop procedural and fluency skills.“The elevation of New Orleans, Louisiana, is 7 feet below sea level. The elevation of Coachella, California, is -72 feet. Write an inequality to compare the two cities.” (6.NS.7 a & b) • Unit 8, Lesson 8, Statistics, Problem Set Guidance, students use what they have learned about mean and median and apply it to describe the center, spread, and overall shape of data. The Problem Set Guidance is as follows: “If a new student walked into our class, how many pockets might the new student be wearing? Which mathematical measure might be the best one to use for such a prediction? Explain your answer. Create a new set of data, different from your class, that has the same mean and median as your class data.” (6.SP.2, 6.SP.5.d) Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example: • Unit 2, Unit Rates and Percent, Lesson 5, Target Task, students engage in procedural fluency and application as they solve, “Market Place is selling chicken for$4.50 per pound. Stop and Buy is selling 5 pounds of chicken for $23.75. You need to buy 8 pounds of chicken. At these rates, which store is cheaper? How much cheaper is it?” (6.RP.3) • Unit 4, Rational Numbers, Lesson 2, Target Task, students develop procedural skills and fluency while demonstrating conceptual understanding: “Write the integer that describes each of the following situations. Then represent the integer on a horizontal or vertical number line. Include the value 0 on your number line and use an appropriate scale. a. A deposit made of$15, b. A withdrawal of $75, c. A credit of$110,  d. A temperature of 15 degrees below 0.”  (6.NS.5)
• Unit 6, Equations and Inequalities, Lesson 2, “Define a solution to an equation as the value of the variable that, when substituted in, makes the equation a true statement.” Students develop procedural skill and fluency, and conceptual understanding through application as they test solutions using substitution and begin to translate situations to equations. Problem Set Guidance is as follows:  “Ana is saving to buy a bicycle that costs $135. She has saved$98 and wants to know how much more money she needs to buy the bicycle. The equation 135 = x + 98 models this situation, where x represents the additional amount of money Ana needs to buy the bicycle. When substituting for x, which value(s), if any, from the set Grade 6, Mathematics Sample, ER Item Claim 2, Version 1.0 {0, 37, 98, 135, 233} will make the equation true? Explain what this means in terms of the amount of money needed and the cost of the bicycle.” (6.EE.5)

#### Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations for practice-content connections. The materials identify, use the Mathematical Practices (MPs) to enrich grade-level content, provide students with opportunities to meet the full intent of the eight MPs, and attend to the specialized language of mathematics.

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The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

All Standards for Mathematical Practice are clearly identified throughout the materials in numerous places, including:

• Each Unit Summary contains descriptions of how the Standards for Mathematical Practices are addressed and what mathematically proficient students should do. An example is the Unit Summary for Unit 3, Multi-digit and Fraction Computation, “By examining the structure of concrete models and patterns that emerge from these structures, students make sense of concepts such as multiplying by a reciprocal of a fraction when dividing or using long division as a shorthand to partial quotients (MP.8).”
• Lessons usually include indications of Mathematical Practices (MPs) within a lesson in one or more of the following sections: Criteria for Success, Tips for Teachers, or Anchor Problems Notes.  An example in Unit 1, Understanding and Representing Ratios, Lesson 2, Criteria For Success is as follows: “Use drawings of ratios as a tool to better understand how two quantities are associated with each other (MP.1).” Lesson 4, Anchor Problem 2: “Continue to monitor student responses for accurate use of language as they describe ratios and their process of determining equivalent ratios (MP.6).” Another example is in Lesson 9, Tips for Teachers: “Students engage in MP.1 in this three-act task as they analyze the information given to them and determine how they can use equivalent ratios to fix the mix-up. They must map out their own strategy and check their answers, making adjustments as needed. Students also discover how they can apply ratio reasoning to support them in understanding the math in the problem and determine a solution (MP.4).”
• In some Problem Set Guidances, MPs are identified within the problem. An example is in Unit 6, Equations and Inequalities, Lesson 3, MARS Formative Assessment Lesson. It states, “This lesson also relates to all the Standards for Mathematical Practice, with a particular emphasis on: 2. Reason abstractly and quantitatively, 4. Model with mathematics, and 7. Look for and make use of structure”

Evidence that the MPs are used to enrich (are connected to) the mathematical content:

• MP1 is connected to mathematical content in Unit 7, Geometry, Lesson 6, as students analyze diagrams to make sense of them in context and determine the math strategies they can use in their solution. They determine any missing measurements they may need and add labels or markings to the diagrams as needed, organizing their work along the way. An example is in the Tips for Teachers section.  It is as follows: “Finally, students should ask themselves at the end if their answer makes sense for the context (MP.1).” Anchor Problem 1, “A carpenter is building a new wall for a house that he is renovating. He knows that there will be a door and a window in the wall. Around the door and window, he uses wooden board to create the wall. A blueprint of the wall is shown below. How much wooden board, in square feet, does the carpenter need to build the wall? Explain your reasoning.”
• MP4: In Unit 2, Unit Rates and Percent, Lesson 5, Criteria for Success, students apply the mathematics they know “to model and solve complex problems involving rate.” “Chris and David run along a bike path toward a pond. Anchor Problem is as follows:  “Chris and David run along a bike path toward a pond. Chris can run 3 miles in 30 minutes, and David can run 5 miles in 60 minutes. They both start running at the same time at the start of the bike path, shown below. a. If both Chris and David run at their current rates, how long will it take each one to get to the pond?  b. Who will be closer to the pond after 90 minutes? How much farther ahead will this person be in front of the other person?” Students have opportunities to take different approaches, organize and explain their strategies so that others, who may have taken a different approach, can follow their line of thinking.
• MP8 is connected to mathematical content in Unit 3, Multi-Digit and Fraction Computation, Lesson 4,  in Anchor Problem 1. It states, “The number 3 is divided by unit fractions 1/2, 1/3, 1/4, and 1/5. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart and answer the questions that follow. What pattern do you notice? What generalization can you make? Explain your reasoning. Notes: For the multiplication problem, students may think of 1/2 × ?=3. This is not incorrect, as it is the related multiplication problem of the division problem shown. However, the focus of this problem is observing the pattern when dividing by a unit fraction; specifically, 3 × 2=? (MP.8).”

There is no evidence where MPs are addressed separately from the grade-level content.

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Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Materials attend to the full meaning of each of the 8 MPs. The MPs are discussed in both the Unit and Lesson Summaries, as appropriate, when they relate to the overall work. They are also explained within specific Anchor Problem notes. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:

MP.1: Students make sense of problems and persevere to a solution.

• Unit 2, Unit Rates and Percent, Lesson 1, Anchor Problem 1 recommends, “Rather than a teacher-led problem, this is a good opportunity to have students work in small groups and determine which strategy they would like to use. Groups can compare different strategies, and the class can discuss which approach they think is best.” The problem is as follows: “Chichén Itzá was a Mayan city in what is now Mexico. The picture below shows El Castillo, also known as the pyramid of Kukulcán, which is a pyramid located in the ruins of Chichén Itzá. The temple at the top of the pyramid is approximately 24 meters above the ground, and there are 91 steps leading up to the temple. How high above the ground would you be if you were standing on the 50th step? The 33rd step? The 80th step?”
• Unit 7, Geometry, Lesson 17, Target Task, “Kelly has a rectangular fish aquarium that measures 18 inches long, 8 inches wide, and 12 inches tall. What is the maximum amount of water the aquarium can hold? If Kelly wanted to put a protective covering over the four glass walls and top of the aquarium, how much material will the cover need.” This problem encourages students to make sense of the problem as they conceptualize volume and area through sketching and labeling the dimensions of the aquarium with the appropriate measurements to answer the questions.

MP.2: Students reason abstractly and quantitatively.

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Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Math FishTank Grade 6 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others through Guiding Questions and Teacher Notes. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 5, Anchor Problem 1 is as follows: “A restaurant that specializes in making pancakes makes 1 batch of pancakes using a ratio of 2 cups of flour to 3 cups of milk. How much flour and milk will the restaurant use to make 2 batches of pancakes? To make 3 batches? Show your reasoning using a visual representation of your choice. On a busy Saturday, the restaurant uses 36 cups of milk for pancakes. How much flour does the restaurant use for the pancakes, and how many batches is this? Show your reasoning using a visual representation of your choice.” Teachers are instructed to guide students in constructing viable arguments with the following Guiding Questions, “What visual representations could you use to represent the ratios in this problem? Which of these representations are reasonable to use for part (a)? Why? Which of these representations are reasonable to use for part (b)? Why? Are there any representations that would work well for both parts of the problem? For part (a) but not (b)? If the restaurant made 7 batches of pancakes on Sunday, how much milk and flour did the restaurant use? How does your visual representation help you see this?”
• Unit 4, Rational Numbers, Lesson 1, Anchor Problem 1, teachers are prompted to facilitate a discussion between students. “An extension of this problem could have students working in pairs, where one student makes a claim similar to Andrea, and the other student agrees or disagrees and explains his or her reasoning.”
• Unit 4, Rational Numbers, Lesson 9, Notes for Anchor Problem 3 is as follows: “This is a good opportunity to have students work in pairs, perhaps after some initial independent time to determine their own responses. Students should defend their reasoning for choosing sometimes, always, or never, using counterexamples where relevant.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 8, Problem 3, teachers are prompted to allow time for students to share their solutions and explain their reasoning: “Are the two expressions below equivalent?” Guiding Questions: a. “Do the variables x and y represent the same number?” b. “Draw tape diagrams for the expressions to see which are equivalent.” c. “How can you use substitution to determine or verify your answer?” Notes: “This is a good opportunity for students to use counterexamples in their explanations to show the two expressions are not equivalent.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 3, teachers are prompted to have students review four expressions written on the board to determine which are correct. Guiding Questions provided are: a. “How did Sam think about the perimeter?” b. “Where did he get the 2?” c. “How did Joanna think about perimeter?” d. “How is it different from Sam?” e. How did Kiyo think about perimeter?” f. “How did Erica think about perimeter?” g. “Whose thinking was she close to and why? Students analyze each of the expressions to understand how each one may have been created...Share and discuss students’ analyses of the expressions so they may hear various arguments from the class.”
• Unit 7, Geometry, Lesson 1, Anchor Problem 1, teachers are prompted to allow time for students to share their thinking to their solution to an area problem. There are guiding questions to ask students that support critiquing the work that they did in class. “Have students discuss in pairs first to compare responses. If students disagree with which parallelograms are marked correctly, have each student explain his or her thinking and justify his or her reasoning. Use the guiding questions to prompt student thinking.”
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Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations that materials use accurate mathematical terminology.

The Match Fishtank Grade 6 materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

Vocabulary is introduced at the Unit Level and reinforced through a Vocabulary Glossary and in the Criteria For Success. For example:

• A Vocabulary Glossary is provided in the Course Summary and lists all the vocabulary terms and examples. There is also a link to the vocabulary glossary on the Unit Overview page for teachers to access.
• Each Unit Overview also has a chart with an illustration that models for the teacher the key vocabulary used throughout the unit.
• Each Unit has a vocabulary list with the terms and notation that students learn or use in the unit. For example, in Unit 1, Understanding and Representing Ratios’ vocabulary includes the following words: ratio, part to part ratio, part to whole ratio, multiplicative relationship, ratio table, double number line, equivalent ratio and tape diagram.
• Unit 4, Rational Numbers, Lesson 9, Criteria For Success, “Define absolute value as the distance from zero on a number line. Understand that absolute value is a distance or magnitude and, therefore, is always positive or zero. Absolute value is never negative.”

Anchor Problem Notes provide specific information about the use of vocabulary and math language (either informal or formal) in the lesson plan. For example:

• Unit 4, Rational Numbers, Lesson 4, Anchor Problem 1 states, “Use this Anchor Problem to introduce and define opposite numbers, including the fact that zero is its own opposite.”
• Unit 7, Geometry, Lesson 8, Anchor Problem 1 is as follows: “When students discuss their strategies in pairs, listen for how students explain their work. How are they describing the shapes they work with?  How are they explaining their process of finding measurements and areas?  Ensure students are accurate and precise in their explanations.”

The Match Fishtank Grade 6 materials support students at the lesson level by providing new vocabulary terms in bold print, and definitions are provided within the sentence where the term is found. Additionally, Anchor Problem Guiding Questions allow students to use new vocabulary in meaningful ways. For example:

• Unit 2, Unit Rates and Percents, Lesson 2, students “Define and understand a rate, associated with a ratio a:b, as a/b units of the first quantity per 1 unit of the second quantity. For example, if a person walks 6 miles in 2 hours, the person is traveling at a rate of 3 miles per hour.”
• Unit 7, Geometry, Lesson 14, Anchor Problem 1 states, “A set of prisms and a set of pyramids are shown. Define and identify face, vertex, edge, and base in the various prisms and pyramids.”

### Usability

##### Gateway 3
Partially Meets Expectations

#### Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for use and design to facilitate student learning. Overall, the design of the materials balances problems and exercises, has an intentional sequence, expects a variety in what students produce, uses manipulatives as faithful representations of mathematical objects, and engage students thoughtfully with mathematics.

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The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials for Match Fishtank Grade 6 meet the expectations that there is a clear distinction between problems and exercises in the materials.

There are eight units in each grade level. Each unit presents lessons in a consistent structure. During the Anchor Problems, which include guided instruction, step-by step procedures, and problem solving, students work on examples and problems to learn new concepts. Problem Set Guidance contains a variety of exercises that allow students to independently master, and demonstrate their understanding of the material. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, MARS Formative Assessment Lessons, Robert Kaplinsky, Yummy Math, EngageNY - Great Minds, and others. Each lesson concludes with a Target Task intended for formative assessment. For Example:

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1, students describe the association between two groups of shapes using ratios. In the Problem Set Guidance (Open Up Resources Grade 6 Unit 2 Practice Problems, Lesson 1, Problems 1-3), students are given additional practice with this skill. In Problem 3, they are shown a picture of two animals: “Write two different sentences that use ratios to describe the number of eyes and legs in this picture.”
• Unit 2, Unit Rates and Percent, Lesson 3, Anchor Problem 3, students find unit rates and use them to solve problems.  An example is as follows: “Emeline can type 2 pages in 8 minutes. What are two rates associated with this ratio? Emeline has a deadline in 18 minutes, at which point she needs to be done typing an article. What is the greatest number of pages that Emeline can type before her deadline? Emeline has to type a 7-page article. How much time will it take her?"  The Guiding Questions for teachers help to lead the student through the problem-solving process.
• Unit 3, Multi-Digit and Fraction Computation, Lesson 3, Anchor Problem 1, students use a visual model to solve a problem that involves dividing a whole number by a fraction. In the Problem Set Guidance, (Illustrative Mathematics Dividing by One-Half ), students are given additional practice with that skill.  An example is as follows: “Solve the four problems below. Which of the following problems can be solved by finding 3÷1/2?”
• Unit 4, Rational Numbers, Lesson 4, Anchor Problem 1, students define and label opposite numbers on a number line through direct instruction and guiding questions.
• Unit 7, Geometry, Lesson 2, Anchor Problems, students  find the area of right triangles, and are introduced to the general formula for the area of a triangle. Students continue to practice this skill in the Problem Set Guidance.
• Unit 8, Statistics, Lesson 3, Anchor Problem 3, students create histograms from tally charts to represent and interpret a data set. In the Problem Set Guidance: (EngageNY Mathematics Grade 6 Mathematics, Module 6, Topic A, Lesson 4, Exit Ticket) additional problems allow for independent practice and mastery,
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Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations that the design of assignments is intentional and not haphazard.

The lessons follow a logical, consistent format that intentionally sequence assignments,  provide for a natural progression, and lead to full understanding for students. For example:

• In the Anchor Problems, students are introduced to concepts and procedures through a problem-based situation. They are guided through the problem solving process via a series of Guiding Questions provided for teachers.
• In the Problem Set Guidance: This portion of instruction connects to the problem learned previously and is the substance of the lesson. It includes a list of suggested resources (including links to resources) or problem types for teachers to create a problem set aligned to the objective of the lesson.  Teachers are encouraged to create a set of problems that best work for the needs of their students or for that particular lesson.
• Each lesson concludes with an independent Target Task designed to cover key concepts from the lesson and formatively assess student understanding and mastery.
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There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials for Match Fishtank Grade 6 meet the expectations that the instructional materials prompt students to produce content in a variety of ways. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 2: Students use double number lines to represent ratios and identify equivalent ratios.
• Unit 3, Multi-Digit and Fraction Computation, Lesson 2: Students use visual models to divide whole numbers by fractions.
• Unit 4, Rational Numbers, Lesson 6: Students use a number line to explain the order of rational numbers.
• Unit 6, Equations and Inequalities, Lesson 1: Students use tape diagrams and balances to represent and solve equations.
##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The series includes a variety of suggestions for physical manipulatives, and links to physical, as virtual manipulatives, for example:

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1 uses a ratio shapes handout allowing students to cut out and sort/group shapes as they describe ratios of groups of objects.
• Unit 3, Multi-Digit Fraction Computation, Lesson 14, Anchor Problem 3  and Problem Set Guidance both use domino cards and a “Factor Game" to teach and reinforce prime factorization. The game is connected to written methods as students transition to writing numbers as a product of prime factors.
• Unit 8, Statistics, Lesson 3, Problem Set Guidance, links to StatKey Descriptive Statistics for One Quantitative Variable to create virtual histograms from data sets.
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials for Match Fishtank Grade 6 are not distracting or chaotic and support students in engaging thoughtfully with the subject.

The entire digital series follows a consistent format, making it easy to follow. The page layouts in the Problem Set Guidance materials are user-friendly, and the pages are not overcrowded or hard to read. Because teachers pre-select material from the suggested sources, they are printed for the students, making it easier to navigate. Graphics promote understanding of the mathematics being learned. The digital format is easy to navigate and is engaging. There is ample space for students to write answers in the student pages and on the assessments.

#### Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials for Match Fishtank Grade 6 partially meet expectations that materials support teacher learning and understanding of the standards. The materials provide questions that support teachers to deliver quality instruction, and the teacher edition is easy to use, consistently organized, and annotated, and explains the role of grade-level mathematics of the overall mathematics curriculum. The instructional materials do not meet expectations in providing adult level explanations of the more advanced mathematical concepts so that teachers can improve their own knowledge of the subject.

##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide teachers with quality questions for students.

In the materials, facilitator notes for each lesson include questions that are provided in Anchor Problem notes for the teacher to guide students' mathematical development and to elicit students' understanding. The materials indicate that questions provided are intended to provoke thinking and provide facilitation through the mathematical practices, as well as, influencing the students to think through their work. For example:

• Unit 2, Unit Rates and Percent, Lesson 4, Anchor Problem and Guiding Questions 2: “Are you able to compare the two options just using the price? Why or why not? Are you able to compare the two options just using the pounds of honey? Why or why not? Why would the unit rate of cost per pound be a useful tool to compare? How can you find the unit rate for each option? What does each unit rate tell you about the options?”
• Unit 4, Rational Numbers, Lesson 3, Anchor Problem and Guiding Questions 2: “What does zero feet mean in this situation? Did the two friends travel in the same or different directions? How do you know? Sketch a number line to represent the elevations of the friends.”
• Unit 7, Geometry, Lesson 12. Anchor Problem 1 and Guiding Questions : “What is the absolute value of each x−coordinate? Of each y−coordinate? What does it mean if two points are symmetrical about an axis? How far is each point from the axis? Would the points (−2,5) and (2,5) line up vertically or horizontally? Which axis would you fold along to match (−2,5) with (2,5)?”
• Unit 8, Statistics, Lesson 4, Anchor Problem 1 and Guiding Questions: “How are the first two graphs similar? How are they different from the third graph? Which graph would you describe as symmetrical? Why? What features make it symmetrical? A skewed distribution has values that are not typical of the rest of the data. These skewed data points can be on the low or the high end. Which graph would you say is skewed left (is skewed toward the smaller values or has a “tail” to the left)? Which graph would you say is skewed right (is skewed toward the larger values or has a “tail” to the right)?”
##### Indicator {{'3g' | indicatorName}}
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations for containing a Teacher Edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials also include teacher guidance on the use of embedded technology to support and enhance student learning.

Tips for Teachers section is present in all lessons, and supports teachers with resources, and an overview of the lesson. These Tips for Teachers sections provide some guidance on how to present the content. For example:

• Unit 3, Multi-Digit Fractions and Computation, Lesson 4, Tips of Teachers: In this lesson, students use visual models and patterns to develop the general rule for dividing by fractions. The focus of this lesson is on the development of the rule rather than on the use of it. Students will have opportunities for a lot of practice in upcoming lessons.

There is guidance for teachers in the form of Guiding Questions and Notes to use when implementing Anchor Problems, Problem Set Guidance, and Target Tasks. Problem Set Guidance sections are optional problem sets within student-facing materials. There is no student edition, so guidance for ancillary materials is not needed.

##### Indicator {{'3h' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials for Match Fishtank Grade 6 do not meet the expectations that materials contain adult-level explanations so that teachers can improve their own knowledge.

There is an Intellectual Prep which includes suggestions on how to prepare to teach the unit; however, these suggestions do not support teachers in understanding the advanced mathematical concepts.

• The teacher materials include links to teacher resources, but linked resources do not add to teacher understanding of the material.
• The materials list Anchor Problems and Target Tasks and provide answers and sample answers to problems and exercises presented to students; however, there are no adult-level explanations to build understanding of the mathematics in the tasks.
##### Indicator {{'3i' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The instructional materials for Match Fishtank Grade 6 meet expectations as they explain the role of the grade-level mathematics in the context of the overall mathematics curriculum.

• Each unit opens with a Unit Summary that includes a Lesson Map and a list of grade-level standards addressed within the unit along with future connections.
• Each Lesson provides current standards addressed in the lesson.
• Each Lesson provides foundational standards which are standards that were covered in previous units or grades that are important background for the current lesson.
##### Indicator {{'3j' | indicatorName}}
Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials for Match Fishtank Grades 6 meet the expectations that materials cross-reference standards and provide a pacing guide.

The Course Summary includes a Pacing Guide. The Pacing Guide does not reference the standards covered but does provide an overview of the number of days expected per Unit. The standards are cross-referenced in multiple places including the Unit Summary, the Lesson Map, Vocabulary, Standards, Mathematical Practices, and Essential Understandings for the Unit. The Lesson provides the objectives, standards, and criteria for success.

##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials for Match Fishtank Grades 6 do not contain strategies for informing parents or caregivers about the mathematics program or give suggestions for how they can help support student progress and achievement.

##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The instructional materials for Match Fishtank Grade 6 include explanations of the instructional approaches of the program. However, there is no identification of research-based strategies.

The Teacher Tools include several handouts that address the instructional approach of the program. An example is as follows: “Components of a Math Lesson (Grades 6-12)”. In addition, there are handouts regarding several instructional strategies. An example is as follows: “A Guide to Academic Discourse” and “A Guide to Supporting English Learners”. The strategies are not identified as research-based.

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials for Match Fishtank Grade 6 partially meet the expectations for offering teachers resources and tools to collect ongoing data about student progress on the CCSSM. The instructional materials provide strategies for gathering information about students’ prior knowledge and strategies for teachers to identify and address common student errors and misconceptions. The assessments do not clearly denote which standards are being emphasized.

##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The instructional materials for Match Fishtank Grades 6 do not meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

There are no diagnostic or readiness assessments, or tasks to ascertain student prior knowledge.

##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide strategies for teachers to identify and address common student errors and misconceptions.

The materials provide notes for teacher guidance for the Anchor Problems that addresses common misconceptions. For example:

• Unit 3, Multi-Digit and Fractional Computation, Lesson 3, Anchor Problem 3 states “There are often common misconceptions around how to treat any remaining fractions of the wholes. In this case, we can count four 2/3 cups in the visual, and we see that there is one third of a cup left over. However, we are counting in servings of 2/3 cup, not 1 cup, so the remaining piece represents 1/2 of a 2/3 cup serving. If students struggle with this concept, there is an additional problem from Illustrative Mathematics’ “Cup of Rice” referenced in the Problem Set Guidance that addresses this further.”
• Unit 6, Equations and Inequalities, Lesson 5, Anchor Problem 2: “A common misconception with problems involving division is for students to divide to find the variable instead of multiply, especially in a problem like the one above where 10 is divisible by 5. Use the diagram to visually represent x as the value that is being divided into 5 pieces, where each piece is equal to 10. Substituting the value of 2 into the equation for x is another way to demonstrate it as an incorrect answer.”
##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials for Match Fishtank Grade 6 meet the expectations for the materials to provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Each lesson is designed with teacher-led Anchor Problems, Problem Set Guidance and Target Tasks. The lessons contain multiple opportunities for practice with an assortment of problems. The Anchor Problems provided the teacher with guiding questions and notes in order to provide feedback for students’ learning. For example:

• Unit 2, Unit Rates and Percent, Lesson 11, Anchor Problem 2 , Notes include the following: “This is also a great opportunity to engage students’ number sense skills. Discuss the different ways that students can solve 22/100×150. For example, 22/100 can be rewritten as 11/50 or as 0.22. There is further simplification that can happen with the 150 in the numerator and 100 or 50 in the denominator to simplify the problem a great deal.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 3, Anchor Problem 2 is as follows:  Students write expressions to match several situations. The guiding questions for teachers ask students to use their expressions to solve problems. In Notes, the following is included: “Use this Anchor Problem to introduce students to simple expressions that involve variables to represent quantities (MP.2). If students get stuck, have them try out different values for the variables to see how the quantities interact with each other.”
##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.

The instructional materials for Match Fishtank Grade 6 meet the expectations for the assessments to clearly denote which standards are being emphasized.

Each unit provides an answer key for the Unit Assessment. The answer key provides each item number and the targeted standard. For example:

• Unit 8, Statistics, Assessment Item 5 correlates with 6.SP.2.
• Unit 3, Multi-Digit and Fraction Computation, Assessment Item 6 correlates with 6.NS.3.
##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials for Match Fishtank Grade 6 do not meet expectations for materials with assessments that include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Each Unit provides a Unit Assessment answer key. The answer key includes the correct answer, limited scoring guidance, and no guidance for teachers to interpret student performance. For example:

• Unit 7, Geometry, Assessment Item 5b states, in the “Correct Answer and Scoring Guidance”:
• Solution: “No, I do not agree with the student. Triangle A and Triangle B have the same measurements for the base and height, so they will have the same area. (Or equivalent)
• 1 pt for determining the student’s claim is incorrect
• 1 pt for justification/explanation.”

There is no evidence of rubrics or guidance for teachers in terms of student performance and suggestions for follow-up.

##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

The instructional materials for Match Fishtank Grade 6 do not provide any strategies or resources for students to monitor their own progress.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The instructional materials for Match Fishtank Grade 6 do not meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners and strategies for meeting the needs of a range of learners. The materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations and include extension activities for advanced students, but do not present advanced students with opportunities for problem solving and investigation of mathematics at a deeper level. The instructional materials also suggest support, accommodations, and modifications for English Language Learners and other special populations and provide a balanced portrayal of various demographic and personal characteristics.

##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials for Match Fishtank Grade 6 partially meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The Lesson Structure provides support for sequencing instruction. Each lesson includes a list of key skills and concepts that students should practice. The program overview states that the lesson core consists of Anchor Problems that lend better to whole group instruction, small group guided discovery, or both. The guiding questions help scaffold and/or extend on each Anchor Problem, but there is no instruction for teachers on how to do this or handle student misconceptions.

It is the teacher‘s discretion to decide how to use the suggestions in the Practice Set Guidance. There is little guidance for the teacher to determine what strategies or materials are provided for scaffolding instruction, how scaffolds are presented, if there is an appropriate mix of problems so all students can find an entry point, and how to identify any large-group misconceptions. The Teacher Tools include a webinar entitled, “ Leveraging Open Education Resources”, but there are no strategies for teachers who struggle to choose and implement the correct materials for each student.

Sequencing and scaffolding are built into each lesson so that teachers pose Anchor Problems with increasing complexity leading to a Target Task. However, if students need additional support, there is no guidance for teachers.

##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials for Match Fishtank Grade 6 partially meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

The Tips for Teachers and Anchor Problem Notes include limited strategies to help teachers sequence or scaffold lessons: "Ask students," “Encourage students to look closely,” "Remind students of a definition," or “Point out to students.” Examples include the following:

• Unit 3, Multi-digit and Fraction Computation, Lesson 6, Anchor Problem 1 Notes are as follows: “Encourage students to show and share multiple ways to solve this problem. One solution may include a tape diagram, another a number line approach. You can also reason through this solution using rates and determining that 1/8 of a credit is the same as 30 minutes. Discuss how writing 1/4  with a common denominator to 7/8 can offer new ways to solve the problem.”
• Unit 6, Equations and Inequalities, Lesson 5, Anchor Problem, Notes are as follows: “Ask students how they would read the equation aloud, what operations are involved, how they would solve an equation with that operation, etc. If students struggle with the fraction, try replacing it with a whole number to gain a better understanding of the operations. However, be sure to replace the fraction and discuss the fractional answer.”
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for Match Fishtank, Grade 6 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

Students engage in tasks throughout lessons in the Anchor Problems, Target Tasks, Problem Set Activities, the 3-Act Math Modeling, and the Mathematics Assessment Project activities. They all present multiple entry points for students. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 9, Target Task is as follows : “Fix the Egg Mixup: What I did: 2 eggs, 4 tablespoons flour; What I should have done: 2 eggs, 3 tablespoons flour.” Presenting this scenario without posing a question allows multiple entry points and varied strategies for students.
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials for Match Fishtank Grade 6 partially meet expectations for suggesting options for support, accommodations, and modifications for English Language Learners and other special populations.

ELL students have support to facilitate their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems). The ELL Design is highlighted in the teaching tools document, "A Guide to Supporting English Learners", which includes strategies that are appropriate for all, but no other specific group of learners. There are no general statements about ELL students and other special populations within the units or lessons.

Specific strategies for support, accommodations, and/or modifications are mentioned in "A Guide to Supporting English Learners" that include sensory, graphic, and interactive scaffolding; oral language protocols, which include many cooperative learning strategies, some of which are mentioned in Teacher Notes; and using graphic organizers with empathize on lighter or heavier scaffolding. An example is as follows:  Oral Language Protocols provide structured routines to allow students to master opportunities and acquire academic language. Several structures are provided with an explanation on ways to incorporation them that include Turn and Talk, Simultaneous Round Table, Rally Coach, Talking Chips, Number Heads Together, and Take a Stand. Ways to adapt the lessons or suggestions to incorporate them are not included within lessons, units, or summaries.

There is no support for special populations.

##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for Match Fishtank Grade 6 do not meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth.

There are limited notes/guidance in the instructional materials that provide strategies for advanced students.

##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide a balanced portrayal of various demographic and personal characteristics. For example:

• Different cultural names and situations are represented in the materials, ie., Sorah, John, Alberto, Beth, and Pedro.
• The materials avoid pronouns, referencing a role instead, ie., the banker, a biologist, a scientist, the soccer team.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials for Match Fishtank Grades 6 provide limited opportunities for teachers to use a variety of grouping strategies.

"The Guide to Supporting English Learners" provides cooperative learning and grouping strategies which can be used with all students. However, there are very few strategies mentioned in the instructional materials.  There are are no directions or examples for teachers in the materials to adapt the lessons or suggestions on when and how to incorporate them. For example:

• In Unit 8, Statistics, Lesson 1, includes the following suggestion: “This is a good opportunity to have students work in pairs in order to explain why a question is/is not a statistical question and to listen to other students explain their reasoning as well.”
##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials for Match Fishtank Grades 6 do not encourage teachers to draw upon home language and culture to facilitate learning.

Materials do not encourage teachers to draw upon home language and culture to facilitate learning although strategies are suggested in "The Guide to Supporting English Learners" found at the Teacher Tools link.

#### Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials for Match Fishtank Grade 6 integrate technology in ways that engage students in the mathematics; are web-­based and compatible with multiple internet browsers; include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology; are intended to be easily customized for individual learners; and do not include technology that provides opportunities for teachers and/or students to collaborate with each other.

##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

Digital materials reviewed for Math Fishtank Grade 6 are included as part of the core materials. They are web-­based and compatible with multiple internet browsers, e.g., Internet Explorer, Firefox, Google Chrome, Safari, etc. In addition, materials are “platform neutral,” i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform. Materials allow for the use of tablets and mobile devices including iPads, laptops, Chromebooks, MacBooks, and other devices that connect to the internet with an applicable browser.

##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The materials reviewed for Math Fishtank Grade 6 do not  include opportunities to assess students' mathematical understandings and knowledge of procedural skills using technology.

##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for Match Fishtank Grade 6 do not include opportunities for teachers to personalize learning, including the use of adaptive technologies.

The instructional materials reviewed for Match Fishtank Grade 6 are not customizable for individual learners or users. Suggestions and methods of customization are not provided.

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials reviewed for Match Fishtank Grade 6 incorporate technology that provides opportunities for teachers and/or students to collaborate with each other.

Students and teachers have the opportunity to collaborate using the Applets that are integrated into the lessons during activities.

##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for Match Fishtank Grade 6 integrate technology including interactive tools, virtual manipulatives/objects, and dynamic mathematics software in ways that engage students in the Mathematical Practices (MPs).

Anchor Problems, Problem Set Guidance activities, and Target Tasks can be assigned to small groups or individuals. These activities consistently combine MPs and content.

Teachers and students have access to math tools and virtual manipulatives within a given activity or task, when appropriate. These include GeoGebra, Desmos, and other independent resources. For example:

• Unit 5, Numeric and Algebraic Expressions, Lesson 6, uses the following technology: Students use a Desmos applet to find match expressions for verbal statements.
• Unit 7, Geometry, Lesson 14, uses the following technology: Students have opportunities to use the GeoGebra applet to create  three-dimensional shapes that open up to reveal the net.

## Report Overview

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

#### Math 3-5

The instructional materials reviewed for Match Fishtank Grades 3-5 meet the expectations for alignment. The assessments at all grade levels are focused on grade-level standards, the materials devote at least 65% of class time to major clusters of the grade, and all grades are coherent and consistent with the standards. Grades 3-5 meet expectations for Gateway 2, rigor and mathematical practices. The lessons include conceptual understanding, fluency and procedures, and application, and there is a balance of these aspects for rigor. The Standards for Mathematical Practice (MPs) are identified, used to enrich the learning, and meet the full intent of all eight MPs. Grades 3-5 partially meet the criteria for usability. The materials meet expectations for Use and Design to Facilitate Student Learning, and partially meet exceptions for Planning and Support for Teachers, Assessment, and Differentiation.

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

#### Math 6-8

The instructional materials reviewed for Match Fishtank Grades 6-8 meet the expectations for alignment. The assessments at all grade levels are focused on grade-level standards, the materials devote at least 65% of class time to major clusters of the grade, and all grades are coherent and consistent with the standards. Grades 6-8 meet expectations for Gateway 2, rigor and mathematical practices. The lessons include conceptual understanding, fluency and procedures, and application, and there is a balance of these aspects for rigor. The Standards for Mathematical Practice (MPs) are identified, used to enrich the learning, and meet the full intent of all eight MPs. Grades 6-8 partially meet the criteria for usability. The materials meet expectations for Use and Design to Facilitate Student Learning, and partially meet exceptions for Planning and Support for Teachers, Assessment, and Differentiation.

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

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

### Overall Summary

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