6th Grade - Gateway 2

The instructional materials for Match Fishtank Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

All units begin with a Unit Summary and indicate where conceptual understanding is emphasized, if appropriate. Lessons begin with Anchor Problem(s) that include Guiding Questions designed to help teachers build their students’ conceptual understanding. The instructional materials include problems and questions that develop conceptual understanding throughout the grade level, especially where called for in the standards (6.RP.A, and 6.EE.3). For example: 

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1, students develop conceptual understanding when introduced to ratios through the use of diagrams. An example of this is as follows:  “In a recipe for oatmeal raisin cookies, the ratio of teaspoons of cinnamon to cups of raisins is 4:8. Draw a diagram to represent the quantities, and write two other ratio statements for the situation.” (6.RP.A)
• Unit 1, Understanding and Representing Ratios, Lesson 4, introduces students to the concept of equivalent ratios. An example of this is Anchor Problem 1, “Are Heather and Audrey’s ratios equivalent? Explain how you know.” (6.RP.1)
• Unit 2, Unit Rates and Percent, Lesson 4, Anchor Problem 1, students are given two different prices for jugs of honey.  Anchor Problem 1 states, “Would you rather buy one 5-pound jug of honey for $15.35, or three 1.5-pound bottles of honey for $14.39? Justify your answer.” (6.RP.3) 
• Unit 5, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 1 uses an area model to show the distributive property conceptually. “Two rectangles were combined to create a larger rectangle, as shown below. “Write as many expressions as you can to represent the area of the larger, outer rectangle.” Guiding question: “How do your expressions connect back to the area model?” (6.EE.3)
• Unit 5, Numerical and Algebraic Expressions, Lesson 9, Anchor Problem 2 uses tape diagrams to discover the concept of using the distributive property to produce equivalent expressions, “The tape diagram represents the expression 3x + 4y. Draw a tape diagram that shows twice the value of 3x + 4y.” Guiding Questions, “What does it mean to take “twice the value” of an expression? What does this look like in a diagram? Rearrange your diagram to group together the same values. What property is this? How does grouping your diagram in this way help you write a new expression?” (6.EE.3) 
• Unit 6, Equations and Inequalities, Lesson 13, Anchor Problem, Question 1 states that students relate variables to the coordinate plane. Students use tables to discover relationships between dependent and independent variables and graph them appropriately: “Determine which variable is dependent and which variable is independent. Make a table showing the number of pencils for 3 – 7 packages. Plot the points in the coordinate plane. If Sarah has 168 pencils, how many packages did she purchase?” (6.EE.9)

Grade 6 materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, Illustrative Mathematics, EngageNY, Great Minds, and others. For example: 

• Unit 1, Understanding and Representing Ratios, Lesson 2, Target Task, students are asked to draw a picture and name two ratios for each given situation: “To make papier-mâché paste, mix 2 parts of water with 1 part of flour. A farm is selling 3 pounds of peaches for $5. A person walks 6 miles in 2 hours.” (6.RP.A)
• Unit 3, Multi-digit and Fraction Computation, Lesson 2, Target Task, students use diagrams to develop the concept of division of fractions by whole numbers: “How much lemonade is in each glass? Write a division problem and draw a visual model” (6.NS.1)
• Unit 4, Rational Numbers, Lesson 3, Problems 1 & 2, (Open Up Resources: Grade 6, Unit 7, Lesson 1 ) give students an opportunity to work independently to demonstrate conceptual understanding of rational numbers by answering questions about temperature, elevation, and sea levels and in some cases to represent points on a vertical number line. “Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level. Drag the points marking the mountains and trenches to the vertical number line and answer the questions: a. Which point in the ocean is the lowest in the world? What is its elevation? b. Which mountain is the highest in the world? What is its elevation? c. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0? d. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.” (6.NS.5)
• Unit 5, Numerical and Algebraic Expressions, Lesson 7, Problem Set Guidance, (Open Middle, Equivalent Expressions 1): Students work independently to explore the use of whole numbers to create equivalent expressions: “Using the whole numbers from 1-9 in the boxes below, create two expressions that are equivalent to one another. You can use each whole number at most once.”  (6.EE.3)
• Unit 5, Numerical and Algebraic Expressions, Lesson 9, Target Task, students complete the following: “For each problem, draw a diagram to represent the expression. Then use the diagram to write an equivalent expression.  a. 4(2m + n)  b. 5x + 15.” (6.EE.3)
• Unit 6, Equations and Inequalities, Lesson 8, Target Task, students define and identify solutions to inequalities. Students are given a list of values and asked, “Which of the following values are solutions to the inequality 5x - 8 $$\lesseq$$ 42. Select all that apply.” (6.EE.5)

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

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

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

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

For example, students independently demonstrate fluency:

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

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

• Unit 5, Numerical and Algebraic Expressions, Lesson 9, students generate equivalent expressions. For example, Problem Set Guidance, Open Middle Distributive Property, states, “Fill in the boxes below using the whole numbers 0 through 9 no more than one time each so that you can make a true equation.” (6.EE.4)
• Unit 6, Equations and Inequalities, Lesson 10. Students work toward developing procedural skills in writing inequalities for real-world conditions. Anchor Problem 3 states, “Two similar situations are described below. Situation A: A backpack can hold at most 8 books. Situation B: A backpack can hold at most 8 pounds. Draw a graph for each situation to represent the solution set. Compare and contrast the two graphs.” Problem Set Guidance provides additional practice. “Write an equation to represent each situation and then solve the equation. Andre drinks 15 ounces of water, which is 3/5 of a bottle. How much does the bottle hold? Use x for the number of ounces of water the bottle holds.” (6.EE.8)

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

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

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

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 3, “To make green-colored water, Brian mixes drops of green food dye and cups of water in a ratio of 4:3.  a. Draw a double number line to represent the ratio of drops of green food dye to cups of water. b. Use your double number line to find 2 equivalent ratios. c. Brian’s friend, Evan, uses a ratio of 20 drops of green food dye to 15 cups of water. Will Evan’s water be the same color green as Brian’s? Explain your reasoning.” (6.RP.3) 
• Unit 3, Multi-Digit and Fraction Computation, Lesson 6, Target Task, “You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is  1 1/2 miles away. You are timing your progress and find that you can travel 2/3 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit. Solve the problem with a diagram and explain your answer. Then write and solve an equation and show that it is the same as what you got in your diagram.?” (6.NS.1)
• Unit 6,  Equations and Inequalities, Lesson 3, Anchor Problem 2, “At a market, a farmer sells apples for $1.33 per pound. At the end of a weekend, the farmer made $74.48 from selling apples. Which equation can be used to determine x, the number of pounds of apples the farmer sold over the weekend.” (6.EE.7)
• Unit 6, Equation and Inequalities, Lesson 7, Anchor Problem 3, “The school librarian, Mr. Marker, knows the library has 1,400 books, but he wants to reorganize how the books are displayed on the shelves. Mr. Marker needs to know how many fiction, nonfiction, and resource books are in the library. He knows that there are four times as many fiction books as resource books. There are half as many nonfiction books as fiction books. a. If these are the only types of books in the library, how many of each type of book are in the library? b. Draw a tape diagram to represent the books in the library, and then write and solve an equation to determine how many of each type of book there are in the library.”  (6.EE.6, 6.EE.7) 
• Unit 6, Equations and Inequalities, Lesson 13, Target Task, “Arian wants to save 20% of his paychecks in a savings account. a. Write an equation to represent the amount Arian should save, s from a paycheck in the amount of p dollars.  b. Create a table of values with at least three or four different paycheck amounts.  c. Plot the values in the coordinate plane to show the relationship between the amount Arian saves and the amount Arian earns. d. Explain how you could use your graph to find how much of a $60 paycheck Arian should put into his savings account.” (6.EE.9, 6.RP.3.a)

Examples of non routine application include, but are not limited to real-world context that are unfamiliar, novel, and/or unrehearsed. For example:

• Unit 2, Unit Rates and Percent, Lesson 14, Problem Set Guidance allows students to apply strategies, organize information and their workspace to keep track of their solution pathway. “Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?”  One possible solution to this non-routine problem uses an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster and 6.EE.B. (6.RP.2, 6.RP.3, 6.RP.3.c, 6.RP.3.d)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 5, Anchor Problem 2, Handout 2, students solve and write story problems involving division with fractions. For example, a. “Make up and solve two of your own slicing problems. In Problem A, you should not have any cheese left over, and in Problem B, you must have some cheese left over. b. For each problem, you need to determine how much cheese you start off with: how long is your block of cheese? You also need to say how thick you want the slices of cheese to be—or you can decide how many slices you will need in total. Keep in mind that the thickness of each slice should be between 1/32 and 1/2 inches thick. c. After you create your problems, make a poster showing each problem and its solution. Each solution should include an explanation, at least one calculation, and a diagram.” (6.NS.1)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 6, Anchor Problem 1, students solve problems involving division with fractions. For example, “It requires 3/4 of a credit to play a video game for one minute. Emma has 7/8 credits. Can she play for more or less than one minute? Explain how you know. How long can Emma play the video game with her 7/8 credits? How many different ways can you show the solution?” (6.NS.1)

The instructional materials in Grade 6 provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 18, Target Task, “In a bag of jelly beans, there are purple jelly beans (grape) and red jelly beans (cherry). For every 4 purple jelly beans, there are 7 red jelly beans. There are 902 jelly beans in the bag. How many of each flavor are there? Choose a strategy to solve the problem. Explain why you chose this strategy and how it shows your solution." (6.RP.3)
• Unit 3, Multi-Digit and Fraction Computation, Lesson 3, Target Task, “A jar has 5 tablespoons of honey in it. One serving of honey is 3/4 of a tablespoon. How many servings of honey are in the jar?” (6.NS.1)
• Unit 6, Equations and Inequalities, Lesson 1, Target Task, “Draw a tape diagram or balance for each equation or situation below. a.  You purchase 4 gift cards, each in the same amount. You spend a total of $60. b. x + 6 = 18. c. 6x = 18.” (6.EE.6, 6.EE.7)
• Unit 6, Equations and Inequalities, Lesson 5, Target Task, Problem 2, “Lee filled several jars with 1/4 cup of water in each jar. He used a total of 8 cups of water. Let j represent the number of jars that Lee filled. Write and solve an equation to find out how many jars Lee filled.” (6.EE.7)
• Unit 6, Equations and Inequalities, Lesson 7, Target Task, “A town's total allocation for firefighters’ wages and benefits in a new budget is $600,000. If wages are calculated at $40,000 per firefighter and benefits at $20,000 per firefighter, write an equation whose solution is the number of firefighters the town can employ if they spend their whole budget. Solve the equation.” (6.EE.6, 6.EE.7)

The instructional materials for Match Fishtank Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Many of the lessons incorporate two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding.

There are instances where all three aspects of rigor are present independently throughout the instructional materials. For example:

• Unit 3, Multi-Digit and Fraction Computation, Lesson 2, Anchor Problem 1, students develop conceptual understanding for dividing fractions. “Leonard made 1/4 of a gallon of lemonade and poured all of it into 3 glasses, divided equally. How much lemonade is in each glass? Write a division problem and draw a visual model.” (6.NS.1)
• Unit 4, Rational Numbers, Lesson 8, Anchor Problem 3, students write inequalities to compare rational numbers in real-world contexts to develop procedural and fluency skills.“The elevation of New Orleans, Louisiana, is 7 feet below sea level. The elevation of Coachella, California, is -72 feet. Write an inequality to compare the two cities.” (6.NS.7 a & b)
• Unit 8, Lesson 8, Statistics, Problem Set Guidance, students use what they have learned about mean and median and apply it to describe the center, spread, and overall shape of data. The Problem Set Guidance is as follows:  “If a new student walked into our class, how many pockets might the new student be wearing? Which mathematical measure might be the best one to use for such a prediction? Explain your answer. Create a new set of data, different from your class, that has the same mean and median as your class data.” (6.SP.2, 6.SP.5.d)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• Unit 2, Unit Rates and Percent, Lesson 5, Target Task, students engage in procedural fluency and application as they solve, “Market Place is selling chicken for $4.50 per pound. Stop and Buy is selling 5 pounds of chicken for $23.75. You need to buy 8 pounds of chicken. At these rates, which store is cheaper? How much cheaper is it?” (6.RP.3)
• Unit 4, Rational Numbers, Lesson 2, Target Task, students develop procedural skills and fluency while demonstrating conceptual understanding: “Write the integer that describes each of the following situations. Then represent the integer on a horizontal or vertical number line. Include the value 0 on your number line and use an appropriate scale.  a. A deposit made of $15, b. A withdrawal of $75,  c. A credit of $110,  d. A temperature of 15 degrees below 0.”  (6.NS.5)
• Unit 6, Equations and Inequalities, Lesson 2, “Define a solution to an equation as the value of the variable that, when substituted in, makes the equation a true statement.” Students develop procedural skill and fluency, and conceptual understanding through application as they test solutions using substitution and begin to translate situations to equations. Problem Set Guidance is as follows:  “Ana is saving to buy a bicycle that costs $135. She has saved $98 and wants to know how much more money she needs to buy the bicycle. The equation 135 = x + 98 models this situation, where x represents the additional amount of money Ana needs to buy the bicycle. When substituting for x, which value(s), if any, from the set Grade 6, Mathematics Sample, ER Item Claim 2, Version 1.0 {0, 37, 98, 135, 233} will make the equation true? Explain what this means in terms of the amount of money needed and the cost of the bicycle.” (6.EE.5)

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

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

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

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

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

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

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

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

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

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

MP.2: Students reason abstractly and quantitatively.

• Unit 6, Equations and Inequalities, Lesson 7, Target Task, students “reason abstractly and quantitatively as they use symbols to represent situations, define their variables, and then interpret their numerical solutions in context: “A town's total allocation for firefighters’ wages and benefits in a new budget is $600,000. If wages are calculated at $40,000 per firefighter and benefits at $20,000 per firefighter, write an equation whose solution is the number of firefighters the town can employ if they spend their whole budget.  Solve the equation.”
• Unit 8, Statistics, Lesson 5, Anchor Problem 2, “Students demonstrate an understanding of the mean or average, as well as an understanding of the relationship between the mean and the data values from which it was calculated.” The problem is as follows: “After finding the average or fair share payment for each person, Person E decides to not take the job because he would be making less money. a. If Person E leaves, then what is the new average payment of persons A–D? b. What impact did Person E leaving have on the average payment?” 

MP 4: Students model with mathematics.

• Unit 3, Multi-Digit and Fraction Computation, Lesson 5, Anchor Problems 1 and 2, [Strategic Education Research Partner (SERP), “No Matter How You Slice It”], students model a real-world application using division of fractions. For example, “If you know the length of a block of cheese, can you determine how many slices it can make? Suppose you get a new block and you know how thick you want your slices. What do you need to know in order to tell how many sandwiches you can make?”
• Unit 7, Geometry, Lesson 9, Anchor Problem 2, students model with mathematics by using a 9 x 9 grid to design a garden and while making adjustments as they try to meet the requirements. “You are responsible for a small plot of land that measures 9 ft. x  9 ft. in the community garden. You want to include $$48 ft^2$$ of gardening space, and you want your garden to have a rectangular shape and a triangle shape. Draw a possible plan for your garden on the grid below. Make sure you do not go outside of the 9 ft. x 9 ft. space.”

MP 5: Students use appropriate tools strategically.

• Unit 5, Numerical and Algebraic Expressions, Lesson 5, Anchor Problem 1, “A company hires five people for the same job for one week. The amount that each person is paid for the week is shown in the table below.” (A table is provided.) “Person D states that the payments are not fair since each person is doing the same job and brings the same set of skills to the job. Everyone agrees that they should all get paid the same amount. How much should each person get paid so that everyone gets the same amount? Assume that the company will spend the same amount as it currently is.” Students can use a variety of tools to solve the problem.
• Unit 7, Geometry, Lesson 1, Anchor Problem 1, “A parallelogram is shown below. a. What strategies could you use to find the area of the parallelogram? b. Follow Steps 1- 4 of this GeoGebra applet Area of Parallelogram to explore the area of parallelograms. Try out different parallelograms by moving the red and blue dots. c. In general, how can you find the area of any parallelogram?” Students can choose their own strategy to solve the problem.

MP.6: Students attend to precision.

• Unit 1, Understanding Ratios, Lesson 4, Anchor Problem 2, students express numerical answers with a degree of precision appropriate to the context of problem using the correct symbols: “How can you create ratios equivalent to 5:6? Create equivalent ratios and reason about how you can create one that correctly matches with 54 cashews.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 11, students “Define variables for real-world contexts with precision.” For example, in the Target Task, “Abel runs at a constant rate. The table below shows how far Abel has run after a certain number of hours. Write an expression to represent the number of miles Abel ran after h hours.”

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

• Unit 5, Numerical and Algebraic Expressions, Lesson 2, Anchor Problem 1, “Evaluate the following numerical expressions: a. 2(5+(3)(2)+4)  b. 2((5+3)(2+4))  c. 2(5+3(2+4)). Can the parentheses in any of these expressions be removed without changing the value of the expression?”
• Unit 8, Statistics, Lesson 4, Anchor Problem 1 presents three histograms and students answer the following questions: “Describe the shape of each distribution and explain what it means about the data set. Which graph is skewed left? Skewed right? Symmetrical? If these histograms represented the wages that people at a company earned, which company would you want to work at? Why? (Assume the same scale in each graph.) When explaining their choice for part (b), students use structural features of the distributions in constructing their arguments.”

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

• Unit 6, Equalities and Inequalities, Lesson 6, Anchor Problem 2, students generalize “through repeated reasoning, an equation to represent the relationship between percent, whole, and part: percent x whole = part.”  “30% of what number is 12?” Solve this problem first by drawing a diagram. Then write and solve an equation to verify your solution. a. Does the 12 represent the whole or the part? b. What would a tape diagram look like?  c. What would a double number line look like?  d. How can you use your diagram to find the missing value?  e. What equation can be used to solve percent problems?  f. What does the equation look like for these values?  g. How will you solve the equation?  h. Does your answer match what you found from the diagram?”
• Unit 7, Geometry, Lesson 4, Anchor Problem 1, students “develop the understanding through repeated reasoning that, regardless of the angles in a triangle, if the base and height are the same, then the area of the triangle is the same.” “Four triangles were made on a geoboard. The pegs on a geoboard are equally spaced in the square grid. a. Which triangle has the greatest area?  b. Which triangle has the least area?  c. Do any of the triangles have the same area? Why is that the case?”

The instructional materials reviewed for Match Fish Tank Grade 6 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 1, Understanding and Representing Ratios, Lesson 11, Problem Set Guidance, (EngageNY Mathematics Grade 6 Mathematics, Module 1, Topic B, Lesson 10, Exit Ticket ad Problem Set, Exercise 1) is an example. Students are given a table of Hours Worked v. Pay, which has an error and instructed, “The following tables were made incorrectly. Find the mistakes that were made, create the correct ratio table, and state the ratio that was used to make the correct ratio table.”
• Unit 4, Rational Numbers, Lesson 4, Target Task is as follows: “Jane completes several example problems that ask her to find the opposite of the opposite of a number, and for each example, the result is a positive number.  Jane concludes that when she takes the opposite of the opposite of a number, the result will always be positive. Do you agree with Jane? Use the number line below to support and justify your answer.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 8, Target Task is as follows: “Students were asked to write a pair of equivalent expressions. The work of four students is shown below. Harry ab=a+a3+a+b+b+b, Iris $$3a^2 = 3 \times 3 x\times a \times a$$, Jill a + a + 1 + a + 2 = 3a + 3, Kevin 2a + 3b = 2 + a + 3 + b. Which student(s) wrote an equivalent pair of expressions? Justify your answer.”
• Unit 7, Geometry, Lesson 2, Target Task 2 is as follows: “Dan and Joe are responsible for cutting the grass on the local high school soccer field. Joe draws a diagonal line through the field, as shown in the diagram below, and says that each person is responsible for cutting the grass on one side of the line. Dan says that is not fair because he will have to cut more grass than Joe. Is Dan correct? Why or why not?”

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

• Unit 2, Unit Rates and Percent, Lesson 2, Problem Set Guidance,(Open Up Resources Grade 6 Unit 8 Practice Problems, Lesson 13), is as follows: “When he sorts the class’s scores on the last test, the teacher notices that exactly 12 students scored better than Clare and exactly 12 students scored worse than Clare. Does this mean that Clare’s score on the test is the median? Explain your reasoning.”
• Unit 3, Mulit-digit and Fraction Computation, Lesson 1, Target Task is as follows: “Seventy-two students in the sixth-grade class are going on a field trip to the aquarium. The math teacher writes this division problem to represent how the students will be grouped for the field trip: 72 ÷ 6=? Abe says, ‘This means that there are 6 students in each group.’ Sam says, ‘This means there are 6 groups of students.’ Who is correct? Explain your reasoning and draw a diagram to support your answer.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 10, Problem Set Guidance (Open Up Resources Grade 6 Unit 6 Practice Problems, Lesson 11, Problem 2), is as follows: “Priya rewrites the expression 8y−24 as 8(y−3). Han rewrites 8y−24 as 2(4y−12). Are Priya's and Han's expressions each equivalent to 8y−24? Explain your reasoning.”
• Unit 8, Statistics, Lesson 8, Anchor Problem 2 is as follows: “At the University of North Carolina (UNC) in the mid-1980's, the average starting salary for a Geography major was over $100,000 (equivalent to almost $300,000 today). At that same time, basketball star Michael Jordan was drafted into the NBA with one of the highest salaries in the league. He had just graduated from UNC with a degree in Geography. Explain why the mean is a misleading measure of center to represent the salary of geography students at UNC. What measure of center would better represent the salary of geography students at UNC? Explain your reasoning.”

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

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

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

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

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

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

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

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

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

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

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