7th Grade - Gateway 2

The instructional materials for Match Fishtank Grade 7 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 (7.NS.A, and 7.EE). For example: 

• Unit 2, Operations with Rational Numbers, Lesson 1, in Anchor Problem 3 , students develop conceptual understanding of rational numbers by representing integers on a number line with arrows and use the length of the arrows to understand absolute value. The number line is used to model addition and subtraction with integers to help develop the concept visually: “Jessica says she’s thinking of two numbers. They are 24 units apart on the number line, and they are opposites. What are the two numbers?” (7.NS.1)
• Unit 2, Operations with Rational Numbers, Lesson 5, the Anchor problems continue to guide students to understanding the addition of rational numbers by modeling traveling along a number line. Given several contextual problems, students are asked to model situations and them to explain what happens.  One problems states, “Joshua travels 5 miles west and then 2 miles west, represented by the equation: -5 + (-2) = -7. a) Model this situation on the number line using arrows and explain what each term in the equation represents in the context. b) Explain, using the context of the situation, why adding two negative integers will always give you a smaller negative integer.” (7.NS.1.b and 7.NS.1.d) 
• Unit 3, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 1 presents the expressions “$$a^2-b$$ and (2ab).” Students determine “Is the expression greater when a = -1 and b = 1 or when a = 1 and b = -1?” (7.NS.3 and 7.EE.1)
• Unit 3, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 2 states, “Write an expression for each sequence of operations: Expression 1: Add 3 to x, subtract the result from 1, then double what you have. Expression 2: Add 3 to x, double what you have, then subtract 1 from the result.” “Evaluate each expression for x = 2” develops conceptual understanding through the Guiding Questions: “What part of both expressions will be the same? What part of each expression will be different? What role do parentheses play in these expressions? Are the two expressions equivalent? How do you know?” (7.NS.3 and 7.EE.A)
• Unit 3, Numerical and Algebraic Expressions, Lesson 6, Anchor Problem 2 states, “Subtract: (3x + 5y − 4) − (4x + 11).” The Guiding Question helps develop conceptual understanding. Guiding Question states: “How can you rewrite the problem without parentheses?” (7.EE.1)
• Unit 3, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 2,  students develop conceptual understanding by using area models to learn how to write expressions: “A square fountain area with side length feet is bordered by a single row of square tiles as shown. What are three different ways to represent the number of tiles needed for the border? Show each representation using the diagram.”  (7.EE.2)
• Unit 5, Percent and Scaling, Lesson 2, Anchor Problem 2 states the following:  “According to the U.S. Environmental Protection Agency, in 2013, people in the United States produced about 254 million tons of trash. Approximately 34.3% of this trash was recycled or composted. About how many million tons of trash were recycled or composted in 2013?” Conceptual understanding of the relationship between percent, part and whole is developed through the Guiding Questions: “Make an estimate of the amount of trash that was recycled or composted. What is the part, percent, and whole in the situation? What would a visual representation of this situation look like? What strategy will you use to solve this problem? Compare your solution to your estimate. Does it seem reasonable?” (7.RP.3 and 7.NS.3)
• Unit 6, Geometry, Lesson 10, Anchor Problem 1, students develop conceptual understanding as they use the formula for the area of the circle: "The circumference of a circle is 24π cm. What is the exact area of the circle? Draw a diagram to assist you in solving the problem." (7.G.4)

Grade 7 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, Proportional Relationships, Lesson 5, Problem Set Guidance: (Open Up Resources Grade 7 Unit 2 Practice Problems, Lesson 5, Problem 2): Students independently develop conceptual understanding of equivalent ratios: “In one version of a trail mix, there are 3 cups of peanuts mixed with 2 cups of raisins. In another version of trail mix, there are 4.5 cups of peanuts mixed with 3 cups of raisins. Are the ratios equivalent for the two mixes? Explain your reasoning.” (7.RP.2)
• Unit 2, Operations with Rational Numbers, Lesson 3, Target Task, students are shown how positive and negative numbers relate to zero on a number line by combining opposite quantities.  An example is as follows: “Rami leaves his house to drive to school. After driving west for 8 3/4 miles, he realizes that he forgot his backpack at home. How far and in what direction does Rami have to travel to get back to his house? Represent the situation on a number line, including labels for Rami’s house and school, and arrows to show his trips.” (7.NS.1)
• Unit 2, Operations with Rational Numbers, Lesson 4, Target Task, students use number lines to build their understanding by finding the solution and graphing it on a number line:  “Represent each addition problem on a number line and find each sum. Then choose one problem and write a real-world situation that could be modeled by the problem." (7.NS.1.b and 7.NS.1.d)
• Unit 3, Numerical and Algebraic Expressions, Lesson 4, Target Task, students simplify expressions by combining like terms with both integer and rational coefficients, as well as with two variables. An example is as follows: “The table below includes expressions that are written in expanded form and in factored form. Complete the table. Use a diagram if needed.”  (7.EE.1)
• Unit 8, Probability, Lesson 2, Target Task, students are asked to independently demonstrate conceptual understanding of experimental vs theoretical probability in the following scenario.  “Each of the 20 students in Mr. Anderson’s class flipped a coin ten times and recorded how many times it came out heads. a. How many heads do you think you will see out of ten tosses? b. Would it surprise you to see 4 heads out of ten tosses? Explain why or why not. c. Here are the results for the twenty students in Mr. Anderson’s class. Use this data to estimate the probability of observing 4, 5, or 6 heads in ten tosses of the coin. (It might help to organize the data in a table or in a dot plot first.”) (7.SP.6)

The instructional materials for Match Fishtank Grade 7 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 these skills, for example: 

• Every Unit begins with a Unit Summary, where procedural skills for the content is addressed.
• In each lesson, the Anchor problem 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 develop procedural skill and fluency throughout the grade level. The instructional materials provide opportunities for students to demonstrate procedural skill and fluency independently throughout the grade level, especially where called for by the standards (7.NS.A, 7.EE.1, 7.EE.4a). For example:

• Unit 2, Operations With Rational Numbers, Lesson 16, Anchor Problem 2, students convert rational numbers to decimals using long division, “Write each decimal as a fraction.” Answer choices:  a. 0.35;  b. 1.64;  c.  2.09; d. -3.125. (7.NS.2.d)
• Unit 3, Numerical and Algebraic Expressions, Lesson 6, Anchor Problem 2, students apply properties of operations as strategies to add, subtract, and expand linear expressions with rational coefficients, “Subtract: (3x + 5y − 4) − (4x + 11).”  (7.EE.1)
• Unit 4, Equations and Inequalities, Lesson 4, Anchor Problem 3, students solve equations in the forms px + q = r and p(x + q) = r algebraically, “Solve the equations. Answer choices: a. 12(x - 2) = 72    b. -1/3(x) + 4 = -2   c. 5.6 - 2p = 13.” Guiding questions are as follows:  “What operation will you undo in each equation first? How do you know that you are maintaining balance in each step you take? How can you check your answer at the end to make sure it is a solution?” (7.EE.4.a).
• Unit 4, Equations and Inequalities, Lesson 5 includes a MARS Formative Assessment Lessons for Grade 7 Solving Linear Equations. An example is as follows:  “The numbers 5, 6 and 7 are an example of consecutive numbers, as one number comes after another. Another three consecutive numbers are added together so that the first number, plus two times the second number, plus three times the third number gives the total. Which of these expressions could represent the total?  Check all that apply.   Answer choices: Total = x + 2x + 3x;  Total = x + 2x + 2 + 3x + 6;  Total = x + 2(x +1) + 3 (x + 2);  Total = x + (2x + 1) + (3x + 2).  Explain your answer.”   (7.EE.3 and 7.EE.4.a)

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

• Unit 2, Operations with Rational Numbers, Lesson 6,  in Target Task, Problem 2, students have the opportunity to independently demonstrate procedural skills in addition of rational numbers . They are asked to, “Find the sums.  a. -3 + (-2 1/4);     b. 5.7 + (-12.2);  c. -9 + 3 2/5 ; d. -10/3 + 18;  e. -2 + 3 + (-1) + 5.”  (7.NS.1.d)
• Unit 2, Operations With Rational Numbers, Lesson 9, students use the same procedure for adding and subtracting signed rational numbers as they do when adding and subtracting integers. For example, Anchor Problem 2 states, “Find each sum or difference.” Answer choices:  a. - 40 2/3 -  8 1/2;  b. -9.08 + 16.52 ;  c. 52 + (-15) (7.NS.1.b and 7.NS.1.c)
• Unit 4, Equations and Inequalities, Lesson 7, students use the distributive property to organize information in word problems in order to write and solve equations. For example, Target Task states, “A batch of 8 cookie ice cream sandwiches weighs 1,092 grams. On average, each cookie weighs 12 grams, and the same amount of ice cream is used for each sandwich. A pint of ice cream has approximately 450–460 grams of ice cream. How many pints of ice cream would you need to make another batch of 8 cookie ice cream sandwiches?” (7.EE.3 and 7.EE.4a)
• Unit 4, Equations and Inequalities, Lesson 10, in the Target Task, students are given the opportunity to independently demonstrate procedural skills in solving inequalities: “Match each inequality to one of the solutions. Justify that your solution is correctly matched to your inequality. Inequalities: -10 - 6x  > 26; -10 + 6x > 46. Solutions: x > -6;  x > 6;  x < -6;  x < 6” (7.EE.4.b)
• Unit 8, Probability, Lesson 1, students use probability to describe impossible, unlikely, equally likely or unlikely, likely, or certain. Problem Set Guidance suggests the following: “Decide where each event would be located on the scale above. Place the letter for each event in the appropriate place on the probability scale. Answer choices,  a. You will see a live dinosaur on the way home from school today.  b.  A solid rock dropped in the water will sink.  c. A round disk with one side red and the other side yellow will land yellow side up when flipped.  d. A spinner with four equal parts numbered 1–4 will land on the 4 on the next spin.  e. Your full name will be drawn when a full name is selected randomly from a bag containing the full names of all of the students in your class.  f. A red cube will be drawn when a cube is selected from a bag that has five blue cubes and five red cubes.  g. Tomorrow the temperature outside will be −250 degrees.” (7.SP.7.a)

The instructional materials for Match Fishtank Grade 7 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 on the Target Task, students engage with problems that have real-world contexts and are presented opportunities for application, especially where called for by the standards (7.RP.A, 7.NS.3, 7.EE.3). 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, Illustrative Mathematics, EngageNY, Great Minds, and others. 

Examples of routine application include, but are not limited to:

• Unit 1, Proportional Relationships, Lesson 18, in Problem Set Guidance (Illustrative Mathematics, Cider Versus Juice - Variation 2), students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. Students are given a picture of a gallon size juice and a box of juice with prices, “Assuming you like juice and cider equally, which product is the better deal? Suppose the juice boxes go on sale for $1.79 for the eight 4.23-ounce juice boxes, and the cider goes on sale for $6.50 per gallon. Does this change your decision?” (7.RP.1 and 7.RP.3)
• Unit 3, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 3, students write and evaluate expressions for mathematical and contextual situations. For example, “Three friends went to the movies. Each purchased a medium-sized popcorn for p dollars and a small soft drink for d dollars. The friends paid together with two twenty-dollar bills. a) Write an expression to represent the amount of money the friends get back in change after paying for their snacks. b) How much change will the friends get back if the concession stand charges $6.50 for a medium-sized popcorn and $4.00 for a small soft drink? (7.NS.3, 7.EE.1) 
• Unit 4, Equations and Inequalities, Lesson 5, in the Target Task, students apply knowledge of solving equations using a real-world context: “Mrs. Canale’s class is selling frozen pizzas to earn money for a field trip. For every pizza sold, the class makes $5.35. They have already earned $182.90, but they need $750. How many more pizzas must they sell to earn $750?” (7.EE.3 and 7.EE.4)
• Unit 5, Percent and Scaling, Lesson 10, Anchor Problem 3, students solve percent applications involving simple interest, commissions, and other fees. For example, “Tyler bought two tickets to a basketball game on the website Game Finder. Each ticket cost $65, and the website charged a convenience fee that was a small percent of the ticket cost. If Tyler’s total bill came to $132.60, what percent was the convenience fee?” (7.RP.3)

Examples of non-routine application include, but are not limited to:

• Unit 1, Proportional Relationships, Lesson 12, Target Task, students use different strategies to represent and recognize proportional relationships. For example, “Oscar and Maria each wrote an equation that they felt represented the proportional relationship between distance in kilometers and distance in miles. One entry in the table paired 152 km with 95 miles. If k represents the number of kilometers and m represents the number of miles, who wrote the correct equation that would relate kilometers to miles? Explain why. a) Oscar wrote the equation k = 1.6m, and he said that the unit rate 1.6/1 represents kilometers per mile. b) Maria wrote the equation k = 0.625m, and she said that the unit rate 0.625 represents kilometers per mile. ‘Sketch a graph that represents the correct proportional relationship between kilometers and miles.” (7.RP.1.2.A, 7.RP.1.2.B, 7.RP.1.2.C, 7.RP.1.2.D) 
• Unit 4, Equations and Inequalities, Lesson 5, students solve word problems using equations in the form px + q = r and P (x + q) = r,.  Each of the Anchor Problems is presented in a real-world or mathematical context. Anchor Problem 1 states, “At the candy store, M&Ms and Skittles are sold for $0.50 per ounce. Kevita puts some M&Ms in a bag and then added 8 ounces of Skittles. The total cost for her bag of candy is $6.50. Kevita and Mary write the equation 0.5(x + 8) = 6.50 to represent the situation, where x represents the number of ounces of M&Ms. Kevita says that to solve this equation, you first distribute the 0.5 through the parentheses to get 0.5x + 4 = 6.50. Mary says that to solve this equation, you first divide by 0.5 on both sides to get x + 8 = 13. .Do you agree with either Kevita or Mary? Why? Finish solving the problem to find out how many ounces of M&Ms Kevita put in her bag of candy.” (7.EE.3 and 7.EE.4.A)

The instructional materials for Grade 7 provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example:

• Unit 1, Proportional Relationships, Lesson 5, Anchor Problem 1, students write equations for proportional relationships from word problems. For example, “A repair technician replaces cracked screens on phones. He can replace 5 screens in 3 hours. a) Write an equation you can use to determine how long it takes to replace any number of screens. b) Write an equation you can use to determine how many screens can be replaced in a certain number of hours. c ) Use one of your equations to determine how long it would take to replace 30 screens. d) Use one of your equations to determine how many screens could be replaced in 12 hours.” (7.RP.1.2, 7.RP.1.2.C)
• Unit 2, Operations with Rational Numbers, Lesson 18, Anchor Problem 2 gives students the opportunity to use mathematics in carpentry. Anchor Problem 2 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?” (7.NS.3)
• Unit 4, Equations and Inequalities, Lesson 12, Anchor Problem 1, gives students the opportunity to use mathematics with food. 3-Act Math Task Sweet Snacks. “Show Act 1 video. Ask, ‘What do you notice? What do you wonder?’ Show Act 2 video.  Ask, ‘What new information do you have? What different combinations of Teddy Grahams and Circus Animals can he buy with $20? How many combinations can you find?’  Show Act 3 - the solution. Ask, ‘how many of the combinations shown did you find? Which combinations did you find that are not shown in the solution video? Are there other combinations that work?” (7.EE.3 and 7.EE.4.A)
• Unit 5, Percent and Scaling, Lesson 5, Anchor Problem 2 gives students the opportunity to use mathematics in a restaurant. Anchor Problem 2 states,  “A restaurant raises its prices by 10% to account for rising prices of supplies and ingredients. The restaurant’s signature pasta dish costs $14 before the price increase. What is the new price of the pasta dish?” a. Solve this problem using any strategy.  b. Mariam solves this problem by finding 110% of $14. Explain why Mariam’s strategy is correct. How does this strategy compare to yours?” (7.RP.1.3, 7.EE.1.2) 

The instructional materials for Match Fishtank Grade 7 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 4, Equations and Inequalities, Lesson 3, Anchor Problem 1, students develop conceptual understanding as they work in groups to solve equations and inequalities. Anchor Problem 1 states, “Divide students into five to seven groups, and give each group the introduction and one of the scenarios below. In each group, students should: a) Represent the scenario with a tape diagram and equation. b) Collaborate on a sequence of operations to find the solution.” Problem Introduction, “The Sanchez family just got back from a family vacation. Jon and Ava are summarizing some of the expenses from their family vacation for themselves and their three children, Louie, Missy, and Bonnie.” A chart with the costs of various items is included chart. “Students are given various scenarios. “Scenario 1: During one rainy day on the vacation, the entire family decided to go watch a matinee movie in the morning and a drive-in movie in the evening. The price for a matinee movie in the morning is different than the cost of a drive-in movie in the evening. The tickets for the matinee movie cost $6 each. How much did each person spend that day on movie tickets if the ticket cost for each family member was the same? What was the cost for a ticket for the drive-in movie in the evening? Scenario 2: For dinner one night, the family went to the local pizza parlor. The cost of a soda was $3. If each member of the family had a soda and one slice of pizza, how much did one slice of pizza cost? They summarize the information in a chart and are asked the following questions: “ 1. Determine the cost of 1 airplane ticket, 2 nights at the motel, and 1 evening movie.  2. Determine the cost of 1 t-shirt, 1 ticket to a baseball game, and 2 days of the rental car.” (7.EE.3 & 7.EE.4.a)
• Unit 2, Operations with Rational Numbers, Lesson 9, Anchor Problem 2, students use procedural skills and fluency to “Find each sum or difference. Rewrite subtraction problems into addition problems or use a number line as needed. a. 18 1/5 − (−5),  b.  −40 2/3 − 8 1/2,  c. −9.08+16.52,  d.  −6−(−12),  e. −11 1/6−(−3), f. 52+(−15),  g. -36.125 + (-14.6)” (7.NS.1.c & 7.NS.1.d) 
• Unit 5, Percent and Scaling, Lesson 2, Anchor Problem 1, students apply their knowledge of mathematics to solve: “In a school election, 60 students voted for the next student body president. Aliyah won 45% of the votes. a. Draw a visual representation of the problem. For example, you could draw a tape diagram or a double number line.   b. Determine how many votes Aliyah won. Choose one of the following strategies: table, percent equation, proportion.” (7.RP.3, 7.NS.3)

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, Operations with Rational Numbers, students build conceptual understanding of adding integers and apply in real-world problems.  Students analyze a number line and are prompted to use a number line in the exercises. For example, Lesson 6, 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.” (7.NS.1.b & 7.NS.1.d)
• Unit 2, Operations with Rational Numbers, Lesson 18, students build conceptual understanding of fraction computations and apply them in real-world problems. For example, Anchor Problem 2 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?” (7.NS.3)
• Unit 4, Equations and Inequalities, Lesson 11, students build conceptual understanding of inequalities and apply to real-world problems. For example, Target Task states, “The members of a singing group agree to buy at least 250 tickets for an outside concert. The group buys 20 fewer lawn tickets than balcony tickets. What is the least number of balcony tickets bought? Write and solve an inequality. Explain your answer in the context of the situation.” (7.EE.4.b)
• Unit 6, Geometry, Lesson 6, Target Task, students build conceptual understanding and demonstrate procedural skill as they solve, “Describe the relationship between the circumference of a circle and its diameter. The top of a can of tuna is in the shape of a circle. If the distance around the top is approximately 251.2 mm, what is the diameter of the top of the can of tuna? What is the radius of the top of the can of tuna?” (7.G.4)  

The instructional materials reviewed for Match Fishtank Grade 7 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. For example, Unit 8, Probability, “Students encounter and use a variety of tools including spinners, dice, cards, coins, etc., and organizational tools such as organized lists, tables, and tree diagrams when they study compound probability (MP.5).”
• Lessons usually include indications of Math Practices within a lesson in one or more of the following sections: Criteria for Success, Tips for Teachers, or Anchor Problems Notes. For example, in Unit 1,  Proportional Relationships, Lesson 7, Tips for Teachers, “Students reason abstractly as they represent proportional situations using tables and graphs, and interpret the information to identify the constant of proportionality and write an equation. Given a graph of a proportional relationship, students re-contextualize information represented in coordinate points to explain what (0,0) and (0,r) mean in context of the problem (MP.2).” Lesson 11, Anchor Problem 2, “In solving part (c), students may use their graph to estimate a time, however, they will realize that the graph alone cannot provide an exact answer. Students may use proportional reasoning to determine the exact rate at which the brothers are running or to write an equation to represent the distance each brother travels over time (MP.4).” Lesson 14, Criteria For Success, “Organize information and map out a solution process for a multi-step problem (MP.1).”

In some Problem Set Guidance sections, MPs are identified within the problem. For example, Unit 5, Percent and Scaling, Lesson 2, MARS Formative Assessment Lesson, Ice Cream, “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:

• MP.1 enriches the mathematical content in Unit 5, Lesson 8, Tips for Teachers,  when “Students make sense of the quantities in each problem, how they are related, what they mean in context, etc., and then determine what strategy or approach they will use to solve (MP.1).” Anchor Problem 1 states, “Sarita is collecting signatures to put a question on her town’s voting ballot.  a. So far, she has collected 432 signatures. This is 36% of the number of signatures she needs. How many signatures does Sarita need to collect?  b. After a week, Sarita now has 582 signatures. By what percent did the number of Sarita’s signatures increase over the week?  c. A friend of Sarita’s says he will help her collect signatures by collecting 12% of the number of signatures she needs. How many signatures will the friend collect?”
• MP2 enriches the mathematical content in Unit 1, Lesson 7, Tips for Teachers, as “Students reason abstractly as they represent proportional situations using tables and graphs, and interpret the information to identify the constant of proportionality and write an equation.” Students are “given a graph of a proportional relationship, students re-contextualize information represented in coordinate points to explain what (0,0) and (0,r) mean in context of the problem (MP.2).” Anchor Problem 2 states, “Three toy cars race down a straight path. The distance each car traveled in meters, over time measured in seconds, is shown in the graph below. At what speed is each car traveling? Where can you see each speed in the graph? Explain the significance of the coordinate point (1,r). What do you think the “r” represents?”
• MP4 enriches the mathematical content in Unit 3, Lesson 9, when students use expressions to model the number of tiles needed for the border of a fountain. Anchor Problem 2 states, “A square fountain area with side length feet is bordered by a single row of square tiles as shown. Each different expression represents the border in a different way, yet all expressions are equivalent as a total number of tiles.  What are three different ways to represent the number of tiles needed for the border? Show each representation using the diagram.”

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

The instructional materials reviewed for Fish MatchTank Grade 7 meet expectations for carefully attending 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, Operations with Rational Numbers, Lesson 6, Target Task 1, students use number lines to represent subtraction of two integers to explore the idea that subtraction of a number is the same as adding its opposite. “Students can make sense of this abstract problem by substituting in specific values for p and q.” “Point A is located at -4.5, and point B is 3.25 units from point A. Write two addition equations that could be used to determine the location of point B. Model this on the number line below.”
• Unit 5, Percent and Scaling, Lesson 7, Anchor Problem 2 states, “There were 24 boys and 20 girls in a chess club last year. This year the number of boys increased by 25%, but the number of girls decreased by 10%. Was there an increase or decrease in overall membership? Find the overall percent change in membership of the club.” Notes: “Students need to map out a solution pathway. In order to find the percent change, students will first need to find the new number of boys and girls in the club this year using strategies from Lesson 5. Students may do this efficiently by finding 125% of 24 boys and 90% of 20 girls. Highlight and showcase examples of students using this efficient approach.”

MP.2: Students reason abstractly and quantitatively.

• Unit 1, Proportional Relationships, Lesson 11, Anchor Problem 1, “A proportional relationship is shown in the graph below. 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.” In this problem, “Students “take an abstract graph and apply a context to it that makes sense with the values given. Students ensure the contexts chosen represent true proportional relationships that could be represented with an equation in the form y=kx.” 
• Unit 3, Numerical and Algebraic Expressions, Lesson 3, Anchor Problem 2 states, “What could be the missing dimensions of the rectangular array? Which set of dimensions uses the greatest common factor of the two terms? Factor the expression below by using a rectangular array. Use the greatest common factor as the representation of the width. 24d - 8e + 6” Students “reason quantitatively in this problem considering the expressions as a whole and also the individual relationship between q and r and the rest of the parts of the expression.” 

MP.4: Students model with mathematics.

• Unit 3, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 2, students write and interpret expressions by demonstration three different ways using tiles. “A square fountain area with side length s feet is bordered by a single row of square tiles as shown. What are three different ways to represent the number of tiles needed for the border? Show each representation using the diagram. Anchor Problem Notes state, “Students use expressions to model the number of tiles needed for the border. Each different expression represents the border in a different way, yet all expressions are equivalent as a total number of tiles (MP.4).” 
• Unit 5, Percent and Scaling, Lesson 14, Anchor Problem 1, students apply the mathematics they know and proportional reasoning to model and determine actual measurements by using a scale. “Vincent proposes an idea to the student government at his school to install a basketball hoop along with a court, as shown in the diagram below. The school administration tells Vincent that his plan will be approved if it fits on the empty lot, which measures 25 feet by 75 feet, on the school property. Will the lot be big enough for the court Vincent planned? Justify your answer.”

MP5: Students use tools strategically.

• Unit 3, Numerical and Algebraic Expressions, Lesson 11, Anchor Problem 1 states, “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.” Students work in groups to solve a real-world problem. Teachers are instructed to, “allow students to request any tools they may find valuable to solve their problems.” 
• Unit 5, Percent and Scaling, Lesson 15, Anchor Problem 2 states, “The map below shows Boston Common and Boston Public Garden. Perla walked around the outside perimeter of the Common and the Public Garden. About how far did she walk?”  Students use appropriate tools strategically as they determine the measurement. “Some students may also ask to use string or some other flexible measuring tool (such as Wikki Stix) because of the curved lengths around the Boston Common.”

MP6: Students attend to precision

• Unit 3, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 1 states, “Given the expression: $$(a^2-b)-(2ab)$$.  Is the expression greater when a = -1, b = 1 or when a = 1 and b = -1?  Students must use precision when representing (-1) and substitute it into the expression to ensure it represents multiplication and not subtraction.” 
• Unit 5, Percent and Scaling, Lesson 12, Anchor Problem 2 states, “Five figures are shown below. a) Which figures are scale images of Figure 1? Explain your reasoning why. b) Which figures are not scale images of Figure 1? Explain your reasoning why.” In the Anchor Problem Notes state, “Students can use the grid to find the measurements of the side lengths to support their reasoning with precision. For example, the dimensions in Figure 3 are half the dimensions of Figure 1, which means they are proportional, but in Figure 5, only the width was doubled and the length stayed the same, which means they are not proportional.”

MP.7: Students look for and make use of structure. 

• Unit 2, Operations with Rational Numbers, Lesson 16, Anchor Problem, Students look for structure in the denominator written in powers of 10. “Malia found a "short cut" to find the decimal representation of the fraction 117/250. Rather than use long division, she noticed that because 250 × 4 =1000. This Anchor Problem encourages students to be thoughtful and efficient in their work. By understanding the structure of the denominator written as a power of 10, students can find an alternative approach to long division in converting a fraction to a decimal; students should develop an awareness that using the general approach of long division is not always the best approach and that thinking critically about the fraction at hand may open up easier approaches.” 
• Unit 6, Geometry, Lesson 2, Anchor Problem 2, students use structure when identifying which terms in an algebraic expression are like terms using vertical angles. For example, “In the diagram below, line CD intersects line AB through point E , Ray EF extends from point E. Callie says that angle CEB is vertical to angle AEF. Explain why her reasoning is incorrect and name the angle that is vertical to angle CEB. A common misconception with vertical angles is to choose angles that appear to be vertical but are not created by two intersecting lines. Have students trace over the lines that intersect to support their understanding of the structure of the diagram, and the significance of the intersecting lines in creating the vertical angles.” 

MP.8: Students look for and express regularity in repeated reasoning.

• Unit 4, Equations and Inequalities, Lesson 10, Anchor Problem 1 states, “Students who need support with inequality  −3x < 12,  will make a chart of values substituting x values into other inequalities so students can see the repeated results as that make an inequality true.”  For example, “In the chart below, one equation and three inequalities are shown. For each one, write the solution and use the space in the last row to check your solution.”
• Unit 6, Geometry, Lesson 6, Anchor Problem 1 states, “Using the circles handout, measure the circumference and diameter of each circle and record your results in the table below. Create a graph to show the relationship between circumference and diameter. Place the values for diameter along the x-axis and the values for circumference along the y-axis. What is the constant of proportionality? Write an equation to relate the circumference and diameter of any circle. This Anchor Problem engages students in the discovery of $$C=\pi d$$ by measuring the circumference and diameter of several circles and, through repeated reasoning, finding the proportional relationship between the two quantities.” 

The instructional materials reviewed for Match Fishtank Grade 7 meet the expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to analyze the arguments of others. 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. For example:

• Unit 2, Operations With Rational  Expressions, Lesson 8, Target Task states, “Two seventh-grade students, Monique and Matt, both solved the following math problem. The temperature drops from 7 °F to -17 °F. By how much did the temperature decrease?  The two students came up with different answers. Monique said the answer is 24 °F, and Matt said the answer is 10 °F. Who is correct? Explain, and support your written response using a number sentence and a vertical line diagram.”
• Unit 3, Numerical and Algebraic Expressions, Lesson 5, Target Task states, “Students in Mr. Jackson’s class are simplifying an expression written on the board: −12p − 5n + 8n + 20. Amal simplified the expression to be 8p + 3n, and Andre simplified the expression to be −32p − 13n. Mr. Jackson saw a few different answers around the classroom, so he gave the students a hint on the board, writing: −12p−5n+8n+20p, −12p+20p−5n+8n; (−12+20)p+(−5+8)n. a. What did Mr. Jackson do in each step he wrote on the board? B. Is either Amal or Andre correct? Explain why.”
• Unit 4, Equations and Inequalities, Lesson 5, Problem Set Guidance, (Illustrative Mathematics Guess My Number), states, “Laila tells Julius to pick a number between one and ten. “Add three to your number and multiply the sum by five,” she tells him. Next she says, “Now take that number and subtract seven from it and tell me the new number.” “Twenty-three,” Julius exclaims. Write an expression that records the operations that Julius used. What was Julius’ original number? In the next round Leila is supposed to pick a number between 1 and 10 and follow the same instructions. She gives her final result as 108. Julius immediately replies: “Hey, you cheated!” What might he mean?”
• Unit 5, Percent and Scaling, Lesson 12 , Target Task states, “Li drew two images of the letter T in a grid, as shown below. Li says the larger T is a scaled image of the smaller T because the larger T is twice the height of the smaller T. Do you agree with Li? Explain why or why not.” 
• Unit 7, Statistics, Lesson 6, Problem Set Guidance, (Open Up Resources Grade 7 Unit 8 Practice Problems, Lesson 15), states, “Clare estimates the students at her school spend an average of 1.2 hours each night doing homework. Priya estimates the students at her school spend an average of 2 hours each night watching TV. Which of these two estimates is likely to be closer to the actual mean value for all the students at their school? Explain your reasoning.”
• Unit 8, Probability, Lesson 6, Target Task states, “Aimee has two sisters in her family. She thinks the probability of a family having three children who are all girls is 1/4 because there can be 0 girls, 1 girl, 2 girls, or 3 girls. Aimee designs a simulation to test her prediction. She flips a coin three times in a row and records the results. She uses heads to represent a girl and tails to represent a boy. After 10 trials of this simulation, Aimee gets the following results.  [a table of the results are given from ten trials] Does 1/4 seem like a reasonable probability of having 3 girls? Explain your reasoning. Estimate the probability of having 3 girls using the results from Aimee’s simulation.”

Student materials consistently prompt students to construct viable arguments. For example:

• Unit 1, Proportional Relationships, Lesson 9, Target Task states, “Is the perimeter of an equilateral triangle proportional to the side length of the triangle? For any regular polygon, is the perimeter of the polygon proportional to the side length of the polygon? Explain your reasoning.” 
• Unit 4, Equations and Inequalities, Lesson  9, Problem Set Guidance, (Open Up Resources Grade 7 Unit 6 Practice Problems, Lesson 13, Problem 2) “How are the solutions to the inequality -3x ≥ 18 different from the solutions to -3x > 18? Explain your reasoning.”
• Unit 6, Geometry, Lesson 3, Problem Set Guidance, (Open Up Resources Grade 7 Unit 7 Practice Problems), Lesson 5, Problem 3: “If you knew that two angles were complementary and were given the measure of one of those angles, would you be able to find the measure of the other angle? Explain your reasoning.”
• Unit 8, Probability, Lesson 1, Problem Set Guidance, (Open Up Resources Grade 7 Unit 8 Practice Problems), Lesson 2, Problem 1, “The likelihood that Han makes a free throw in basketball is 60%. The likelihood that he makes a 3-point shot is 0.345. Which event is more likely, Han making a free throw or making a 3-point shot? Explain your reasoning.”
• Unit 8, Probability, Lesson 5, Problem Set Guidance, (EngageNY Mathematics, Grade 7, Mathematics, Module 5, Topic B, Lesson 10), Problem Set, Question 1c: “How do the simulated probabilities in part (b) compare to the theoretical probabilities of part (a)?”

The instructional materials reviewed for Math FishTank Grade 7 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, Proportional Relationships, Lesson 3, Anchor Problem 2, teachers are prompted to encourage students to construct viable arguments when determining if a proportional relationship exists between two objects. Guiding Questions: a. “Why might one of these hoses not have a proportional relationship between time and the amount of water?” b. “If a relationship is not proportional, can you determine other missing values in the relationship?” c. “Why or why not?” Notes: “This is a good opportunity for students to construct arguments and to hear and critique the arguments of others.” 
• Unit 2, Operations with Rational Numbers, Lesson 8, Anchor Problem 3 states, “Darnell thinks that –4 is less than –6 because 4 is smaller than 6, and –4 is closer to 0 than –6 is. Draw a number line to show the numbers 0, –4, and –6. Then explain why Darnell is incorrect.” The following Guiding Questions are provided to assist teachers in analyzing the reasoning of Darnell, “What happens to the size of numbers when you move to the right on a number line? To the left? What integer is directly to the right of 0? Directly to the left? Are –4 and –6 to the right or left of 0 on the number line? Which number, –4 or –6, is farther to the left of 0? What is an example of a number that is less than –6? Greater than –4?”
• Unit 3, Numerical and Algebraic Expressions, Lesson 7, Anchor Problem 1, Guiding Questions are provided to support students in constructing a viable argument and analyzing the arguments of others. Example is as follows:  “What action did each student (Len, Monty, Nailany) take in their first step to simplify the expression? For the students who made a misstep, how would you explain it to them? How might Oscar’s rewritten expression help Len understand what to do? What is the expression written in simplified form?”
• Unit 6, Geometry, Lesson 5, Anchor Problem 3 states, “Hilary is ordering a large circular pizza from her local pizza shop. She asks about the size of the pizza and is given 3 different values: 14 inches, 44 inches, and 7 inches. Hilary knows three measurements that are used to describe circles: radius, diameter, and circumference. Which value best matches each measurement? Explain your reasoning.” Teachers are given the following Guiding Questions to engage students in constructing a viable argument:  a. “Circumference is similar to perimeter of polygons in that it is the distance around a circle. What part of the pizza best relates to the circumference?”  b. “Is the radius smaller or greater than the diameter?”  c. “Do you think the circumference is smaller or greater than the diameter? Why?” In the Notes, teachers are given suggestions to engage student in analyzing the reasoning of others: “This is a good opportunity to pair up students who arrive at different answers. Each student can defend his or her matches and try to convince their peers of their answer. “
• Unit 6, Geometry, Lesson 15, Anchor Problem 1 states, “This is a good opportunity for students to construct arguments to defend their determinations, and to proactively seek any counterexamples that may be used to show more than one triangle can be created.” 
• Unit 8, Probability, Lesson 4, Anchor Problem 3 states, “A six-sided number cube includes the numbers 1, 3, 5, and 7, as shown below (1 and 5 appear twice). Han predicts that out of 150 rolls of the number cube, the number 1 will appear exactly 50 times. Do you agree with Han? Explain your reasoning.” Guiding Questions are provided for teachers to engage students in critiquing the arguments of others: a. “Is this a fair number cube?” b. “What is a reasonable low number of times for the number 1 to appear over 150 rolls?” c. “What is a reasonable high number of times for the number 1 to appear over 150 rolls?” d. “How can you adjust Han’s claim to make it more reasonable?” e. “What other predictions can you make about the number cube being rolled?”

The instructional materials reviewed for Match Fishtank Grade 7 meet the expectations that materials use accurate mathematical terminology.

The Match Fishtank Grade 7 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. It is 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 4, Equations and Inequalities' vocabulary includes the following words: equation, solution, inequality, substitution, and tape diagram.
• Unit 2, Operations with Rational Numbers, Lesson 16, Criteria For Success, “Define terminating decimals as numbers with a finite number of digits after the decimal point, and define repeating decimals as numbers with an infinite number of digits after the decimal point in which a digit or group of digits repeats indefinitely.”

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 2, Operations with Rational Numbers,  Lesson 5, Anchor Problem 3 states, “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. Listen for students describing the absolute value of numbers in their consideration of the signs of the sums.”
• Unit 7, Statistics, Lesson 2, Anchor Problem 2 states, “Use this Anchor Problem to introduce the concept of random sampling, in which every person in the population has an equal chance of being chosen for a sample. In discussing these methods, students should construct arguments as to why method E is the only one that would not be biased or unrepresentative.”

The Match Fishtank Grade 7 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 4, Equations and Inequalities, Lesson 1, the term “solution” is in bold print, and the definition is provided within the sentence: “Define and understand a solution to an equation as a number for a variable that makes the equation a true statement when substituted in.”
• Unit 5, Percent and Scaling, Lesson 10, Anchor Problem 1, “What is interest? Explain in your own words. If you were to graph the relationship between time and interest, what shape would the graph be?”
• Unit 6, Geometry, Lesson 5, Anchor Problem 1 states, “Draw a circle in two ways. Based on your drawings, how would you define a circle? 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.”