The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. According to Course Summary, Learn More About Fishtank Math, Our Approach, “Procedural Fluency AND Conceptual Understanding: We believe that knowing ‘how’ to solve a problem is not enough; students must also know ‘why’ mathematical procedures and concepts exist.” Each lesson begins with Anchor Problems and Guiding Questions, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. Examples include: 

• In Unit 3, Numerical and Algebraic Expressions, Lesson 1, Anchor Problem 1, students develop conceptual understanding as they use order of operations to evaluate numerical expressions.It states, “Evaluate the following numerical expressions. −2(5 + (3)(−2) + 4); −2((5 + 3)(−2 + 4)); −2(5 + 3(−2 + 4)). Can the parentheses in any of these expressions be removed without changing the value of the expression?” The following Guiding Questions support discourse and the development of conceptual understanding, “Describe what is happening in each expression. What role do the parentheses play in each expression? How do they change the way you work with each expression? How is the 3 treated differently in the third expression than in the second expression? What is the order of operations? How does it guide you in evaluating expressions to the correct answer? How does it help you in this problem?” (7.EE.1, 7.NS.3)

• In Unit 4, Equations and Inequalities, Lesson 2, Anchor Problem 2, students develop conceptual understanding of two-step equations, representing them in the forms px + q = r and p(x + q) = r using tape diagrams. It states, “Which equation matches the tape diagram shown below? For the other equation, draw a tape diagram to represent it. Equations: Equation A: 3x + 4 = 45 Equation B: 3(x + 4) = 45.” A tape diagram is shown. A rectangle is divided into 3 equal sections. Each section is labeled x + 4 and the total of the three pieces is labeled 45. The following Guiding Questions support discourse and the development of conceptual understanding, “What role do the parentheses play? How do they make the equations different? What story situation could the tape diagram represent? How does a tape diagram help you visualize a situation and the equation it represents? How does a tape diagram help you understand how to solve the equation?” (7.EE.4a)

• In Unit 6, Geometry, Lesson 5, Anchor Problem 1, students define a circle and identify the measurements of radius, diameter, and circumference. It states, “Draw a circle in two ways: Method 1: Draw a point on a blank piece of paper and label it P. Using a ruler, measure exactly 2 inches from point P and make a point. Measure exactly 2 inches from point P in a different direction and make a point. Continue measuring and adding points until the shape of a circle starts to appear. Method 2: Draw a point on a blank piece of paper and label it Q. Using a compass, set the measurement to $1\frac{1}{2}$ inches. Draw a circle using the compass. Based on your drawings, how would you define a circle?” The following Guiding Questions support discourse and the development of conceptual understanding, “What is the importance of the center point of a circle? How is a circle created? How are the two methods similar? How are they different? What are characteristics of a circle? What is the relationship between any point on the circle and the center point of the circle?” Teacher notes state, “Use this Anchor Problem to define a circle as a closed shape defined by the set of all points that are the same distance from the center point of the circle. Students likely have not used compasses before and may need additional support using this tool.” (7.G.4)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Problem Sets, or student practice problems, can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding of key concepts, are designed for independent completion. Both problem types, when appropriate, provide opportunities for students to independently demonstrate conceptual understanding. Examples include:

• In Unit 1, Proportional Relationships, Lesson 1, Problem Set, Problem 4, students solve ratio and rate problems by using double number lines, tables, and unit rate. It states, “A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour. Complete the table as you answer the questions that follow.” A table is shown with honey (tbsp) in the x column and flour (c) in the y column and the following coordinates: (8,10), (20,__ ), (13,__ ), and (__, 20). Students respond to the following: “a. How many cups of flour do they use with 20 tablespoons of honey? b. How many cups of flour do they use with 13 tablespoons of honey? c. How many tablespoons of honey do they use with 20 cups of flour? d. In your own words, explain how you came up with the solution to part (c).” (7.RP.1)

• In Unit 2, Operations with Rational Numbers, Lesson 1, Target Task, students demonstrate understanding of opposites and absolute value by representing rational numbers on a number line. It states, “Point A is shown on the number line below.” Students are provided a number line with a rational number positioned between 0 and -1. Students respond to the following questions: “a. What number does point A represent? b. What is the absolute value of the number represented by point A? c. What is the opposite of the number represented by point A? Indicate this on the number line. d. What is the distance between point A and -1 on the number line?” (7.NS.1) 

• In Unit 8, Probability, Lesson 4, Problem Set, Problem 5, students reason about theoretical probability. It states, “At a store, if a customer makes a purchase, they can spin a wheel to see if they win a small gift. The wheel is shown below.” A wheel with six equal-sized spaces includes: Not a winner, $5 gift card, Not a winner, Free pencil, Free tote bag, Free pencil. Students respond to the following questions: “a. If the store has 500 customers, about how many $5 gift cards should be they prepared to give away. b. The store manager expects there to be about 600 customers over the weekend. She reasons that she will need exactly 200 pencils. Do you agree with the manager’s reasoning? Would you recommend she have more or less pencils ready for the weekend? Explain your reasoning.” (7.SP.6, 7.RP.3)

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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

• In Unit 1, Proportional Relationships, Lesson 3, Anchor Problem 1, students practice calculations with proportional relationships. The problem states, “Randy is driving from New Jersey to Florida. Every time Randy stops for gas, he records the distance he traveled in miles and the total number of gallons of gas he used. a. Assume that the number of miles driven is proportional to the number of gallons of gas used. Complete the table with the missing values. b. If the relationship between gallons of gas and miles was not proportional, could you still complete the table? Explain why or why not.” Students are provided a table with gallons of gas used and miles driven. (7.RP.2) 

• In Unit 2, Operations with Rational Numbers, Lesson 16, Anchor Problems, students convert rational numbers. The problem states, “1. Write each decimal as a fraction: a. 0.35, b. 1.64, c. 2.09, d. -3.125. 2. Write each rational number as a decimal. a. $$-\frac{5}{6}$$, b. $$\frac{9}{16}$$, c. $$-\frac{7}{9}$$, d. $$\frac{25}{12}$$. Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “How do you read the decimal “0.35”? What is the place value of the number and what does that tell you about the denominator when it’s written as a fraction? Can any of the fractions from question 1 be written in an equivalent, simpler way? What is a general strategy for converting a decimal to a fraction? Which decimal values in question 2 will be negative? Positive? How can you use long division to convert from a fraction to a decimal? What happens in a long division problem when a decimal terminates? What happens in a long division problem when a decimal repeats? How do you know if your decimal is a repeating decimal?” (7.NS.2d)

• In Unit 2, Operations with Rational Numbers, Unit Summary, describes how students build fluency within seventh grade units and build into eighth grade expectations. It states, “In several upcoming units, seventh-grade students will rely on their increased number sense and ability to compute with rational numbers, in particular in Unit 3, Numerical and Algebraic Expressions, and in Unit 4, Equations and Inequalities. By the time students enter eighth grade, students should have a strong grasp on operating with rational numbers, which will be an underlying skill in many algebraic concepts. In eighth grade, students are introduced to irrational numbers, rounding out their understanding of the real number system before learning about complex numbers in high school. Included in the materials for this unit are some activities that aim to support and build students’ fluency with integer computations, especially mental math.” (7.NS.A)

• In Unit 7, Statistics, Lesson 8, Anchor Problem 3, students compare and make inferences about populations. The problem states, “James wants to get to work as quickly and reliably as possible in the mornings. He tries three different transport methods: cycle all the way; drive all the way; walk to the railway station, take the train, and walk from the station. He tries each method several times and records how many minutes the entire journey takes.” A table is shown with 10 entries for the bicycle route, eight entries for the car route, and six entries for the walk-train-walk route. Students respond to the following: “a. Use the data to make a case for why he should travel to work by bicycle. b. Use the data to make a case for why he should travel to work by car.” Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “What do you notice about the times it took James to get to work using each transportation method? Do you notice any outliers? What might have caused those? What information about the data sets will help you compare the transportation methods? What are the measures of center? What are measures of variability?” (7.SP.3, 7.SP.4)

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

• In Unit 4, Equations and Inequalities, Lesson 4, Target Task, students solve equations in the forms px + q = r and p(x + q) = r  algebraically. The task states, “Solve each equation for the variable. Show every step in your work that maintains the balance in each equation. a. $$\frac{1}{2}$$(x + 8) = -10 b. -5x + 12 = 20 c. (set up as balance scale) x + x + x + 1.5 + 1.5 + 1.5 = 1.5 + 1.5 + 1.5 + 1.5” (7.EE.4a)

• In Unit 6, Geometry, Lesson 11, Problem Set, Problem 2, students make calculations with area and circumference of a circle. The problem states, “Complete the following table. Use 3.14 for $$\pi$$. Round to the nearest tenth.” A table is provided with four rows and columns for circumference, radius, area, or diameter. The value of the area or circumference is given and students calculate the other values. (7.G.4, 7.G.6)

• In Unit 8, Probability, Lesson 8, Target Task, students calculate theoretical probability of compound events. The task states, “Lin plays a game that involves a standard number cube and a spinner with four equal sections numbered 1 through 4. If both the cube and spin result in the same number, Lin gets another turn. Otherwise, play continues with the next player. What is the probability that Lin gets another turn?” (7.SP.8)

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Anchor Problems, at the beginning of each lesson, routinely include engaging single and multi-step application problems. Examples include:

• In Unit 2, Operations with Rational Numbers, Lesson 18, Anchor Problem 2, students use four operations to solve routine problems with rational numbers (7.NS.3). The problem states, “Michael’s father bought him a 16-foot board to cut into shelves for his bedroom. Michael plans to cut the board into 11 equal lengths for his shelves. a. The saw blade that Michael will use to cut the board will change the length of the board by -0.125 inches for each cut. How will this affect the total length of the board? b. After making his cuts, what will the exact length, in inches, of each shelf be?” 

• In Unit 3, Numerical and Algebraic Expressions, Lesson 11, Anchor Problem 1, students model a non-routine real world problem using rational numbers and reasoned estimates (7.EE.3). The problem states, “Trophy stand, Central High School won the league softball championship game last weekend. The team would like to make a display stand for the trophy. The stand will be a rectangular prism. The players plan to paint the stand white so they can each paint a handprint on the stand in different colors. The players want to fit each of their handprints on the stand without overlapping with any other handprints. There are 24 players on the team. Design a display stand that meets the requirements above. What are its dimensions? Show or explain how you found the dimensions using words, pictures, and/or numbers.”

• In Unit 5, Percent and Scaling, Lesson 8, Anchor Problem 1, students solve routine percent problems, including percent increase and decrease (7.EE.2, 7.RP.3). The problem states, “Sarita is collecting signatures to put a question on her town’s voting ballot. She has three weeks to collect the signatures. At the end of each week, Sarita finds the total number of signatures she has collected. a. At the end of week 1, Sarita has collected 528 signatures. This is 44% of the number of signatures she needs. How many signatures does Sarita need to collect? b. At the end of week 2, Sarita has a new total of 704 signatures. By what percent did the number of Sarita’s signatures increase over the week? c. At the end of week 3, her total number of signatures increased by 75%. Does Sarita have enough signatures? Justify your answer.”

• In Unit 6, Geometry, Lesson 7, Anchor Problem 2, students solve a non-routine circumference problem (7.G.4). The problem states, “Two figures are shown below. Figure A is a semi-circle, and Figure B is composed of a square and two semi-circles. Find the distance around each figure.” Figure A is shown with a diameter of 5ft and Figure B is shown with the square’s side length of 5ft. 

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

• In Unit 1, Proportional Relationships, Lesson 5, Problem Set, EngageNY Mathematics, Grade 7 Mathematics > Module 1 > Topic B > Lesson 9, students use the constant of proportionality to represent proportional relationships with equations in real world routine contexts (7.RP.2, 7.RP.2.c).  Example 1, Problem Set 1 and 3 state, “Jackson and his grandfather constructed a model for a birdhouse. Many of their neighbors offered to buy the birdhouses. Jackson decided that building birdhouses could help him earn money for his summer camp,but he is not sure how long it will take him to finish all of the requests for birdhouses. If Jackson can build 7 birdhouses in 5 hours, write an equation that will allow Jackson to calculate the time it will take him to build any given number of birdhouses, assuming he works at a constant rate. a. Write an equation that you could use to find out how long it will take him to build any number of birdhouses. b. How many birdhouses can Jackson build in 40 hours? c. How long will it take Jackson to build 35 birdhouses? Use the equation from part (a) to solve the problem. d. How long will it take to build 71 birdhouses? Use the equation from part (a) to solve the problem.”

• In Unit 4, Equations and Inequalities, Lesson 5, Target Task, students write and solve an equation for a routine real-world problem (7.EE.3). The task states, “Giselle’s youth club sells cookies to fund their trips and activities. Each year they need to earn $1,247 from selling cookies. For each box of cookies they sell, they make $1.45. If Giselle’s club has already made $472.70 from selling cookies, how many more boxes do they need to sell to meet their goal?”

• In Unit 6, Geometry, Lesson 9, Problem Set, Problem 3, students solve a non-routine real-world problem using the relationship between the area of a circle and its diameter (7.G.4). The problem states, “Morgan painted a small circle on her paper. She thinks that if she paints a second circle with twice the diameter as the first one, that she’ll need twice as much paint. Is Morgan right? Use an example to explain your answer.” 

• In Unit 7, Statistics, Lesson 6, Problem Set, Problem 5, students analyze the impact of sample size on variability and the accuracy of predictions for a non-routine problem (7.SP.2). The problem states, “Why does a greater variability tend to lead to a lower accuracy of predictions?”

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout Grade 7. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 1, Proportional Relationships, Lesson 15, Problem Set, Problem 1, students develop procedural skill and fluency as they use proportions to solve rate and ratio problems. The problem states, “Solve the following proportions. a. $$\frac{4}{5}=\frac{x}{8}$$, b. $$\frac{7}{2}=\frac{14}{x}$$, c. $$\frac{m}{4}=\frac{5}{10}$$, d. $$\frac{y}{3}=\frac{10}{6}$$, e. $$\frac{10}{8}=\frac{11}{w}$$, f. $$\frac{1}{6}=\frac{x}{12}$$.” (7.RP.1: Compute unit rates associated with ratios of fractions, and 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems.) 

• In Unit 2, Operations with Rational Numbers, Lesson 4, Anchor Problem 3, students extend their conceptual understanding of positive and negative integers by using a number line to model addition problems. The problem states, “In part (a), model the addition problem on the number line to find the sum. In part (b), write an addition equation to represent what is shown on the number line. a. 5+(−4)+(−3).” Part b shows three arrows above a number line, one starting at 5 and pointing left, another starting at -1 and pointing right, the third with a closed circle at 0 and pointing left. (7.NS.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtract on a horizontal or vertical number line diagram.) 

• In Unit 6, Geometry, Lesson 16, Target Task, students apply their understanding of geometrical figures as they describe two-dimensional figures that result from slicing three-dimensional figures.The task states,  “A cube is sliced with a single straight cut, creating a two-dimensional cross-section. Name 2 different two-dimensional shapes that could result from the slice, and explain or draw how they are created.” (7.G.3: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Examples include:

• In Unit 3, Numerical and Algebraic Expressions, Lesson 9, Target Task, students develop all three aspects of rigor simultaneously, conceptual understanding, procedural skill and fluency, and application, as they write and interpret expressions. The task states, “A picture with dimensions x by y inches, is framed by a rectangular border. The border is 1 inch wide, as shown in the figure below. Write 2 different expressions to represent the area of the border around the picture.” The figure shown has an inner rectangle with length, y, and width, x, and an additional rectangle surrounding the first, to represent the frame. (7.EE.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.)

• In Unit 5, Percent and Scaling, Lesson 13, Anchor Problem 1, students develop conceptual understanding alongside procedural skill and fluency as they define and determine the scale factor between two images. The problem states, “Pentagon FGHIJ is a scale image of pentagon ABCDE. a. Complete the table below with the measurements of each pentagon. b. Is there a proportional relationship between the side lengths of the original pentagon and the scaled pentagon? Explain using the measurements in the table. c. What is the constant of proportionality? d. What is the scale factor?” An image of two pentagons and a chart is provided for students. Students use the pentagons and fill in the chart with Original line segment and Corresponding line segment lengths. (7.G.1: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale, and 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems.)

• In Unit 7, Statistics, Lesson 4, Problem Set, Problem 3, students develop conceptual understanding alongside application as they analyze data using measures of center and interquartile range. The problem states, “Your school has a small bookstore where you can buy school supplies such as pens, pencils, and notebooks. The table below shows the number of notebooks purchased on Mondays and Fridays over 2 months. a. Find the mean number of notebooks purchased at the school store on Mondays and the mean number of notebooks purchased on Fridays. b. Find the median number of notebooks purchased at the school store on Mondays and the median number purchased on Fridays. c. Which measure of center, the mean or the median, best represents the typical number of notebooks purchased on Mondays? On Fridays? Explain your reasoning for each choice.” (7.SP.3: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability, and 7.SP.4: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.)

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

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

• In Unit 1, Proportional Relationships, Lesson 14, students find the unit rate and use it to solve problems. Anchor Problem 1 states, “A small pool is leaking water from a hole. After $$2\frac{1}{2}$$ minutes $$3\frac{1}{2}$$ liters of water have leaked out. a. At this rate, how many liters will have leaked out after 10 minutes?  b. If the pool has 42 liters of water in it, after how many minutes will it be empty?” Teachers are provided the Guiding Questions, including, “What are the two unit rates that you could find for this ratio? Which one is useful to solve part (a)? Explain what each unit rate means in this context. How can you use the unit rate of liters per minute to solve part (b)?” Notes after the guiding questions provide additional guidance for teachers, including, “Students may use a few different strategies here. Some students may set up a complex fraction to find the unit rate of water leaking per minute and then multiply by the number of minutes. Some other students may set up a table to help guide them through the calculations.“ A Criteria for Success for this lesson states, “Organize information and map out a solution process for a multi-step problem (MP.1).” 

• In Unit 2, Operations with Rational Numbers, Lesson 6, students add and reason about sums of rational numbers. Anchor Problem 3 states, “The temperature of water in a lake at 9:00 a.m. is -2.6˚C. By noon, the temperature of the water rises by 5.1˚C. By 9:00 p.m., the temperature of the water falls by 12.8˚C from what it had been at noon. Write an addition problem to represent the changing temperature of the water, and find the temperature of the water at 9:00 p.m.” One Criteria for Success for this lesson states, “Make sense of problems using understanding of distance, order, magnitude, direction, etc. (MP.1).” Guiding Questions provide additional guidance for teachers, including, “Before you compute the answer, will the temperature at the end of the day be positive or negative? How do you know? What strategy will you use to add three addends? Is there more than one way to approach this problem?” 

• In Unit 4, Equations and Inequalities, Lesson 2, students represent equations in the forms px + q = r and p (x + q) = r and use tape diagrams to solve problems. Target Task states, “A family of 5 went to a matinee movie on a Saturday afternoon. The movie tickets for the matinee were a special price for each person. The family spent a combined $25 at the concession stand on drinks and popcorn. Altogether, the family spent $57.50 at the movies. a. Draw a tape diagram to represent the situation above. Then write an equation. b. How does the tape diagram help you understand how much the special price was?” A Criteria for Success for this lesson states that students, “Understand how a tape diagram can be helpful to visualize a solution pathway for an equation (MP.1).” 

• In Unit 6, Geometry, Lesson 10, students solve problems involving area and circumference of two-dimensional figures. Problem Set, Problem 4 states, ”A circular pathway is 157 feet around. Inside the pathway is a grass field. What is the area of the grass region inside the pathway?” Tips for Teachers state, “Students engage in MP.1 and MP.7 as they make sense of complex geometric figures, looking for significance of shapes and measurements within the structure of the diagram, and looking for entry points to solve the problem. Support students by asking questions about what they observe in the diagrams and what initial strategy ideas they have.” 

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 1, Proportional Relationships, Lesson 11, students make connections between the four representations of proportional relationships. Anchor Problem 1 states, “A proportional relationship is shown in the graph below (graph is pictured). a. Describe a situation that could be represented with this graph. b. Write an equation for the relationship. Explain what each part of the equation represents.” Guiding Questions include additional guidance, “What are some examples of proportional relationships in real life? What is the unit rate in this graph? Label the axes and title the graph to represent your situation. What does the point (0, 0) represent in your situation? Use your equation to determine some examples for your situation.” Notes for the teacher state, “Students take an abstract graph and apply a context to it that makes sense with the values given (MP.2). Ensure the contexts chosen represent true proportional relationships that could be represented with an equation in the form y = kx.” 

• In Unit 5, Percent and Scaling, Lesson 7, students find the percent of increase or decrease given the original and new amounts. Target Task states, “In April, Justin sent 675 text messages on his phone. In May, he sent 621 text messages. By what percent did the number of text messages Justin sent decrease from April to May?” Tips for Teachers state, “Students continue to reason abstractly, making meaning of the quantities in the problems to understand their relationships before doing any calculations (MP.2).” 

• In Unit 7, Statistics, Lesson 5, students analyze sets of data and make a recommendation based on that analysis. Problem Set, Problem 4 states, “Justin is interested in joining a soccer team on the weekends, but he wants to determine if the time commitment is right for him. He looks at two different soccer teams, the Rapids and the Timbers. Data for each team is shown below. The Rapids had a mean of 98 minutes spent in practice and at games on the weekend, with a MAD of 45 minutes. The Timbers had a mean of 125 minutes spent in practice and at games on the weekend, with a MAD of 15 minutes. Justin has a few other activities on the weekends, so he needs to have a consistent schedule without committing too much time to soccer. Which soccer team would you recommend to Justin? Why?” The Criteria for Success provides additional guidance for teachers, “Compare data sets with similar or the same mean but different MADs and draw inferences related to the context of the problem (MP.2).” 

• In Unit 8, Probability, Lesson 3, students reason about the probability in a real world problem. Problem Set, Problem 4 state, “A school is rewarding students who have perfect attendance with a chance to win a gift card to a local bookstore. The office manager determines that there are 14 sixth grade students, 10 seventh grade students, and 6 eighth grade students who have perfect attendance and will be entered into the raffle. a. What is the probability that a 7th grade student is chosen? b. The office manager realizes that some eighth graders were accidently not included. After these eighth grade students’ names are added to the raffle, the probability that a seventh grade student is chosen is now $$\frac{5}{18}$$. How many eighth grade students were added to the raffle?”

he materials reviewed for Fishtank Plus Math Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP3 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes) and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 4, Equations and Inequalities, Lesson 9, students construct viable arguments and critique the reasoning of others as they solve inequalities. Anchor Problem 2 states, “Identical copies of books are being packaged and shipped to a bookstore. There are 18 boxes of books that each weigh 31 pounds. The boxes have space for some more books to be added; however, the total weight of all of the boxes cannot exceed 615 pounds. Two workers, Kevin and Olivia, want to determine how many more pounds of books they can add into each box without going over the weight limit. Kevin writes the inequality 18(x + 31) ≥ 615. Olivia writes the inequality 18(x + 31) ≤ 615. Who wrote the correct inequality? How many extra pounds can they add to each box?” Guiding Questions provide support to the teacher for development of MP3, “What does each part of the inequality represent? Should the weight of the books be more than 615 pounds or less than 615 pounds? Which inequality symbol represents this? Predict a value that you feel confident is a solution. Test it in each inequality to see which one makes a true statement. If each book weighs 1 pound, how many books can be added to each box?” Problem Notes state, “If there are mixed responses from students, this would be a good opportunity to have students from each position argue why he/she thinks Kevin or Olivia is correct, and then have a student from the opposing position respond. As a class, students can decide whose argument is valid (MP.3).”

• In Unit 5, Percent and Scaling, Unit Assessment, students construct viable arguments and critique the reasoning of others as they calculate and compare simple interest with different rates and investments. Problem 10 states, “Hayden invested $200 in a savings account that earned 3.3% simple interest each year. William invested $300 in a savings account that earned 2.5% simple interest each year. Neither Hayden nor William add or take out money from their accounts. Hayden claims that she will earn more interest on her account than William after 2 years because she has a greater interest rate. Do you agree with Hayden? Why or why not? Be sure to defend your answer with numbers.”

• In Unit 7, Statistics, Lesson 3, students critique the reasoning of others and construct a viable argument as they generate random samples for a statistical question. Problem Set, Problem 3 states, “Suppose 45% of all the students at Andre’s school brought in a can of food to contribute to a canned food drive. Andre picks a representative sample of 25 students from the school and determines the sample’s percentage. He expects the percentage for this sample will be 45%. Do you agree? Explain your reasoning.” The Unit Summary outlines connections to the MP, “Throughout the unit, students reason about data, make connections, and defend their reasoning by constructing arguments (MP.3).”

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

MP4: Model with mathematics, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students are given many opportunities to solve real-world problems, identifying important quantities to make sense of relationships, and representing them mathematically. They model with math as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 2, Operations with Rational Numbers, Lesson 7, Anchor Problem 2, students model subtraction as addition of the opposite value (or additive inverse). The problem states, “Use your number line and game piece to model each subtraction problem and answer the questions that follow. a. 5 - (-3)  b. 3 - (-5)  c, -3 - (-5) d, -5 - (-3). 1. What does each part of the subtraction problem tell you to do on the number line? 2. What answer do you get for each subtraction problem? 3. Rewrite each subtraction problem as an addition problem with the same value. Model the addition problem on the number line to show it has the same value. What changes and what stays the same?” One Criteria for Success for this lesson states, “Model addition and subtraction problems on the number line (MP.4).”

• In Unit 4, Equations and Inequalities, Lesson 3, Target Task, students model real-world problems with tape diagrams and equations. The task states, “Riley takes two walks every day, one in the morning and one in the evening, and walks for a total of $$5\frac{1}{4}$$ hours in a 7 day week. If he walks for 15 minutes each morning, how many minutes does he walk for each evening? Draw a tape diagram and write an equation to represent the situation. Use either model to solve.” Unit 4 Summary states, “Throughout the unit, students encounter word problems and real-world situations, covering the full range of rational numbers, that can be modeled and solved using equations and inequalities (MP.4).”

• In Unit 8, Probability, Lesson 5, Problem Set, Problem 5, students design and conduct simulations to model real-world situations. The problem states, “A store sells spinning toys that come in 6 different designs. Each toy is sold separately, and is wrapped in packaging so that you do not know which design you are buying. You want to know about how many toys you would need to buy before you had one of each design. Design and describe a simulation you could perform to determine an answer to your question.” 

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

• In Unit 1, Proportional Relationships, Lesson 11, Target Task, students use tools to represent proportional relationships. “Patrice makes a spicy salsa by adding red pepper flakes to a chunky tomato mix in proportional amounts. For example, she mixes $$\frac{1}{2}$$ teaspoon of red pepper flakes to 2 cups of tomato mix. Represent the relationship between red pepper flakes, in teaspoons, to tomato mix, in cups, in two different ways (table, graph, or equation). In your work, define any variables that you use.” 

• In Unit 4, Equations and Inequalities, Unit Assessment, Problem 8, students choose from strategies they have learned to solve a real-world problem. “There are three brothers in the Howard family, Daryl, Malik, and Terrance, whose ages add up to 26 years. Daryl is 3 times as old as Malik, and Terrance is 5 years older than Daryl. Use any strategy, such as a tape diagram or equation, to find the age of each brother. Show your work.” 

• In Unit 6, Geometry, Lesson 5, Problem Set, Problem 6, students use radius, diameter and circumference of a circle to solve problems. “Use a mathematical tool to draw a circle with center point A and a radius of 2 inches.” Unit 6 Summary states, “Students should have access to several tools they may opt to use throughout the unit, including rulers, protractors, compasses, and reference sheets (MP.5).”

he materials reviewed for Fishtank Plus Math Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP6 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes), and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

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

• In Unit 2, Operations with Rational Numbers, Lesson 5, students attend to precision as they reason about the sums of rational numbers. Anchor Problem 3 states, “For each problem, determine if the sum will be positive, negative, or zero. You do not need to find the sum. a. 4 + (−1);  b. −14 + (−10); c. 7 + (−8); d. 12.25 + 13.7; e. −22 + 20, f. −5$$\frac{1}{2}$$+ 7; g. −4 + 4; h. −7 + (−7)”. Problem Notes for teachers state, “Listen for precise use of language in students’ responses to the first problem. For example, saying the sum in part (a) is positive because ‘4 is the bigger number and 4 is positive’ is not accurate because the same reasoning does not hold true for part (c) where 7 is the bigger number but the sum is not positive (MP.6). Listen for students describing the absolute value of numbers in their considerations of the signs of the sums.” 

• In Unit 5, Percent and Scaling, Mid-Unit Assessment, students attend to precision when calculating percent decrease. Problem 4 states, “A monthly subscription to Moviez, a media streaming service, costs $12 per month. Moviez also offers a yearly subscription for $110. What is the percent decrease in money spent per year if you change from a monthly to a yearly subscription? Give your answer to the nearest whole percent.” 

• In Unit 7, Statistics, Lesson 5, students attend to precision when calculating and reasoning about mean and mean absolute deviation for a data set. In Problem Set, Problem 1 states, “Over the course of a week, Bryant kept track of the number of ounces of water he drank each day. Here are his results: 44, 56, 65, 60, 40, 55, 58. a. Find the mean number of ounces of water Bryant drank. b. Find the mean absolute deviation (MAD) of the data set. c. The following week, Bryant drinks the same mean number of ounces of water, but he was more consistent with how much water he drank each day. Did the value of the MAD increase or decrease? Explain your reasoning.” 

Students have frequent opportunities to attend to the specialized language of math in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 1, Proportional Relationships, Unit Assessment, students use the specialized language of mathematics as they reason about proportions and explain their thinking. Problem 10 states, “The table below shows the amounts, in ounces, of tomato sauce and cheese used to make the last 4 orders at Sara’s Pizza. a. Is the relationship between number of pizzas and amount of cheese proportional? Explain your reasoning. b. Sara estimates that they will sell approximately 400 pizzas next week. About how much tomato sauce and cheese should she order to be prepared for next week? Show your work clearly.”

• In Unit 3, Numerical and Algebraic Expressions, Lesson 2, students use the specialized language of mathematics as they evaluate algebraic expressions. Anchor Problem 1 states, “Given the expression: $$(a^2-b)-(2ab)$$ Is the expression greater when a=-1, b=1,  or when a=1, b=-1?” A Problem Note for the teacher states, “Take notice of how students are representing (-1) when they substitute it into the expression to ensure it represents multiplication and not subtraction (MP.6).” 

• In Unit 6, Geometry, Lesson 5, students use the specialized language of mathematics as they define the features of a circle and identify measurements related to circumference, radius, and diameter. Target Task states, “The circle below has a center at point A and a radius of 4.5 cm. Which of the line segments below can you determine the measurement of? Find the measurement of the line segments that you can determine. For any one you cannot determine, explain your reasoning why. a. AB, b. AC, c. DF, d. AE, e. DE, f. AD.” Lesson 5, Criteria for Success states, “1. Understand that a circle is a closed shape defined by the set of all points that are the same distance from the center point of the circle. 2. Understand that the radius is the distance or the line segment from the center of the circle to any point on the circle. 3. Understand that the diameter is the distance or the line segment from one point on the circle, through the center, to another point on the circle. 4. Understand that the circumference is the measurement of the distance around the circle.” The Unit 6 Summary states, “Throughout the unit, students encounter several vocabulary words, such as complementary angles, vertical angles, radius, and circumference. Many of these words enable students to be more precise in their communications with each other (MP.6).”

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

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

• In Unit 3, Numerical and Algebraic Expressions, Lesson 7, students make use of structure as they simplify expressions using their knowledge of the properties of operations. In Problem Set, Problem 5 states, “Select all of the expressions that are equivalent to 4.75x − 2(3x − 2.6) − 3.25. A. 4.75x − 6x − 5.2 − 3.25; B. 4.75x − 6x + 1.95;  C. −1.25x + 1.95; D. 4.75x − 6x + 2.6 − 3.25; E. −1.25x + 5.2 − 3.25.” Unit 3 Summary states, “Students pay particular attention to the structure of expressions in order to better understand what an expression means and how it can be manipulated (MP.7).”

• In Unit 4, Equations and Inequalities, Lesson 10, students make use of structure as they reason about inequalities with negative coefficients. Anchor Problem 3 states, “Rakiah is solving the inequality -2x+5>1. She is unsure how to treat the negative 2 in the inequality, so she uses the addition property of equality to rewrite her inequality. See her work below. -2x + 5 + 2x > 1 + 2x, 5 > 2x + 1, 5 - 1 > 2x +1 - 1, 4 > 2x, $\frac{4}{2}$ > $\frac{2x}{2}$, 2 > x Is Rakiah’s solution correct? Justify your answer.” Problem Notes for the teacher state, “This anchor problem uses the structure of the inequality to demonstrate how, when solving inequalities with negative coefficients, it is not the inequality symbol that ‘flips’ but rather the placement of the variable in the problem. By adding 2x to both sides, the variable becomes positive, but it is now read as ‘less than,’ where before, on the left side, it was ‘greater than’ (MP.7).” 

• In Unit 8, Probability, Lesson 7, students make use of structure as they consider the probability of compound events and organize the sample space using different tools. Target Task states, “An experiment involves flipping a fair coin and rolling a fair six-sided die. a. List all possible outcomes of the experiment. Use an organized list, table, or tree diagram. b. What is the probability of getting a head and an even number? c. What is the probability of getting a tail and the number 4? d.What is the probability of getting a 5?” Tips for Teachers state, “Following the introduction to compound events in Lesson 6, in this lesson students learn how to organize the sample space for compound events. Within each organizational tool, there is a structure embedded that ensures each possible outcome is accounted for (MP.7). Ensure students see and can utilize this structure as they create their own organized spaces.” 

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

• In Unit 2, Operations with Rational Numbers, Lesson 14, students use repeated reasoning in order to make generalizations about division with signed numbers. In Problem Set, Problem 1 states, “Determine if the following quotients are positive or negative. You do not need to compute them. a. 7 ÷ -3 b. -8 ÷- 4 c. $\frac{1}{2}$ ÷ -9 d. -(-4 ÷ (-9)) e. 13 ÷ 7 f. -(13 ÷ 7).” A Criteria for Success in the lesson states, “Use the relationship between multiplication and division to determine the rules for dividing signed numbers (MP.8).” 

• In Unit 5, Percent and Scaling, Lesson 17, students use repeated reasoning as they draw scale images and compute areas from scale drawings. Anchor Problem 1 states, “In the grid below, draw a scale image of the rectangle using a scale factor of 2. Then complete the table. Repeat the activity using a scale factor of 3. 1. When the scale factor was 2, how much was the area scaled by? 2. When the scale factor was 3, how much was the area scaled by? 3. If the scale factor of an image is n, how much is the area of the image scaled by?” Problem Notes for teachers state, “Use this Anchor Problem to guide students to the discovery that the area of an image scales by the square of the scale factor. The grids support this concept visually by enabling students to see 4 copies of the original rectangle when the scale is 2, or 9 copies when the scale is 3. This concept is also demonstrated numerically in the table; encourage students to observe and annotate patterns and connections in the table (MP.8).” 

• In Unit 7, Statistics, Lesson 6, students use repeated reasoning to draw conclusions about sampling size, variability, and prediction accuracy. Anchor Problem 1 states, “Students work in groups of two or three. Have half of the class conduct Trial A and the other half of the class conduct Trial B. (Note, Trial B may take slightly longer than Trial A due to the increased sample size.) Trial A: Select 3 cubes from the bag and take note of how many blue cubes are in your sample. Calculate and record the percentage of cubes in your sample that are blue. Repeat the first two steps 10 times. Draw a dot plot to represent the percentages of blue cubes from your 10 samples. Find the mean and mean absolute deviation (MAD) of your data. Make a prediction of the percentage of blue cubes in the bag based on your data. Trial B: Select 10 cubes from the bag and take note of how many blue cubes are in your sample. Calculate and record the percentage of cubes in your sample that are blue. Repeat the first two steps 10 times. Draw a dot plot to represent the percentages of blue cubes from your 10 samples. Find the mean and MAD of your data. Make a prediction of the percentage of blue cubes in the bag based on your data. As a class: a. Share results from each trial as a class (see Notes below for a suggestion). b. Find the actual percentage of blue cubes in the bag. c. How do the mean percentages of blue cubes from Trial A and Trial B compare to the population percentage? Which trial was more accurate? Why do you think this is so? d. What conclusions can you draw about sampling size, variability, and accuracy?” Problem Notes for teachers state, “Before the class discussion, it would be beneficial if the dot plots could be displayed (with each trial grouped together) for the whole class to see. These could be taped around the room and students could do a quick gallery walk around the room to observe patterns, similarities, or differences between the trials across the results of the class (MP.8).”

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

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

• In Unit 4, Equations and Inequalities, Unit Summary provides an overview of content and expectations for the unit. Within Unit Prep, View Unit Launch, there is a video detailing the content for teachers. The materials state, “Welcome to the Unit Launch for 7th Grade Math, Unit 4 Equations and Inequalities. Please watch the video below to get started.” Additionally, the Unit Summary contains Intellectual Prep and Unit Launch with Standards Review, Big ideas, and Content connections. The Standards Review provides teachers an “opportunity to reflect on select standards from the unit. [...] In this section you will examine the language of these standards and reflect on how several problems on the end-of-unit assessment relate to the standard.”  Then, Big Ideas help teachers “understand how these ideas develop throughout the unit by analyzing lessons and problems from the unit, and finally, have the chance to reflect on how you will address your students’ needs around these concepts.” Finally, Content Connections states, “In this final section of the unit launch, you’ll have the chance to zoom out and look at the related content that students study before and after this unit.” This information is included for Units 1-5 in Grade 7.

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

• In Unit 1, Proportional Relationships, Lesson 4, Tips of Teachers provide guidance for teachers to support students in connecting proportional relationships from a table to an equation. The materials state, “Lessons 4 and 5 focus on representing proportional relationships as equations. Equations are abstract and can be challenging for some students to grasp. Encourage students to return to the table to show the relationship between the two quantities, either adding a column to show the constant of proportionality or drawing an arrow across rows and indicating the multiplication. Ensure that students know what the variables in the equation represent to keep the context connected to the abstract form.”

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

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

• In Unit 2, Operations With Rational Numbers, Unit Summary, Intellectual Prep Unit Launch, Content Connections, Future Connections, contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. “Looking ahead, students continue their study of the number system in 8th grade with an introduction to irrational numbers. In studying 8.NS.A, students learn that all rational numbers have a terminating or repeating decimal expansion, and those that do not are considered irrational. Students use their knowledge of rational numbers to estimate and reason about the values of irrational numbers. The problems below are examples of problems from the 8th grade curriculum. Notice how students build upon their prior knowledge of rational numbers from 6th and 7th grades. Students’ abilities to compute with rational numbers will be needed throughout their algebra work in middle school and beyond. Simplifying expressions and solving equations will require students to use both properties of operations and their skills with adding, subtracting, multiplying, and dividing signed rational numbers. The problem below is an example from 8th grade where students solve for the variable. Take note of the skills students need with rational number computation in order to solve. In high school, students will learn additional properties of rational and irrational numbers (HSN.RN.B), and they will learn how computing with polynomials is analogous to computing with rational numbers in that they form a closed system under the four operations (HSA.APR).”

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

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

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

• In 7th Grade Math, Standards Map includes a table with each grade-level unit in columns and aligned grade-level standards in the rows. Teachers can easily identify a unit when each grade-level standard will be addressed. 

• In 7th Grade Math, Unit 5, Percent and Scaling, Lesson Map outlines lessons, aligned standards, and the lesson objective for each unit lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• In Unit 6, Geometry, Lesson 18, the Core Standard is identified as 7.G.B.6. The Foundational Standard is identified as 6.G.A.4. Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Problems, Problem Set, and Target Task. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each Unit Summary includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Additionally, for Units 1 through 5, there is a Unit Launch within the Intellectual Prep that includes more information about content connections. Examples include:

• In Unit 2, Operations with Rational Numbers, Unit Launch, Connections states, “The cluster heading for 7.NS.A says to ‘Apply and extend previous understandings of operations with fractions’. In line with this, students have a lot of prior knowledge around computing with fractions that they bring with them to this unit. Furthermore, students have prior knowledge around rational numbers as they exist on the number line, both to the right and the left of zero. In this 5th grade problem aligned to 5.NF.A.1, students add mixed numbers with different denominators. They are first asked to find an estimate before computing. This type of reasoning will be helpful to students in 7th grade when they work with more complex values, both positive and negative. Looking ahead, students continue their study of the number system in 8th grade with an introduction to irrational numbers. In studying 8.NS.A, students learn that all rational numbers have a terminating or repeating decimal expansion, and those that do not are considered irrational. Students use their knowledge of rational numbers to estimate and reason about the values of irrational numbers.”

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. This information can be found within Our Approach and Math Teacher Tools. Examples where materials explain the instructional approaches of the program include:

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

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

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

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

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

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

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 7th Grade Course Summary, Course Material Overview, Course Material List 7th Grade Mathematics states, “The list below includes the materials used in the 7th grade Fishtank Math course. The quantities reflect the approximate amount of each material that is needed for one class. For more detailed information about the materials, such as any specifications around sizes or colors, etc., refer to each specific unit.” The materials include information about supplies needed to support the instructional activities. Examples include:

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

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

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

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

• Unit 6 uses protractors, one per student.

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

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for having assessment information included in the materials to indicate which standards and mathematical practices are assessed. 

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

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

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

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

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each Post-Unit Assessment Analysis provides an answer key, potential rationales for incorrect answers, and a commentary to support analysis of student thinking. According to Math Teacher Tools, Assessment Resource Collection, “commentaries on each problem include clarity around student expectations, things to look for in student work, and examples of related problems elsewhere on the post-unit assessment to look at simultaneously.” Each Mid-Unit Assessment provides an answer key and a 1-, 2-, 3-, or 4-point rubric. Each Pre-Unit Assessment provides an answer key and guide with a potential course of action to support teacher response to data. Each lesson provides a Target Task with a Mastery Response. According to the Math Teacher Tools, Assessment Overview, “Target Tasks offer opportunities for teachers to gather information about what students know and don’t know while they are still engaged in the content of the unit.” Examples from the assessment system include:

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

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

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

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

The materials reviewed for Fishtank Plus Math Grade 7 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

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

• In Unit 2, Operations with Rational Numbers, Post-Unit Assessment, Problems 8, 10, and 14 develop the full intent of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers). Problem 8 states, “A baker regularly uses flour to make pastries for the bakery. At the beginning of the month, the baker has a 25 pound-bag of flour. Each week she uses $$5\frac{3}{5}$$ pounds of flour. a. Write an expression to represent the change in weight of the bag of flour over 4 weeks. b. How much flour is left in the bag after 4 weeks? Show your work clearly.” Problem 10 states, “The table below shows the weekly change in the price of one gram of gold for four weeks. a. By how much did the price of one gram of gold change from the beginning of week 1 to the end of week 4? Did the price increase or decrease? Show your work clearly. B. At the end of week 4, the price per gram of gold was $39.28. What was the price per gram of gold at the beginning of week 1?” Problem 14 states, “Ava and Jiao each swam a two-lap swimming race. Ava took 31.49 seconds to finish her first lap and 30.03 seconds to finish her second lap. Jiao finished her two-lap swimming race 1.76 seconds faster than Ava. What was Jiao’s total swimming time, in seconds, after she finished her two-lap race?”

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

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

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