8th Grade - Gateway 2

The instructional materials for Match Fishtank Grade 8 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 (8.EE.B, 8.G.A, and 8.F.A). For example: 

• Unit 1, Exponents and Scientific Notation, Lesson 10, Anchor Problem 1 reviews students’ understanding of large numbers and how they compare to one another: “There are about one million people living in Austin, Texas. There are twice as many people living in Houston, Texas. Phil says that means there are one billion people living in Houston. Explain why Phil is incorrect.”  (8.EE.3, 8.EE.4) 
• Unit 1, Exponents and Scientific Notation, Lesson 1, Anchor Problem, Question 1, (from Robert Kaplinsky's: “How Did They Make Ms. Pac-Man”?) students are shown a video of Ms. Pac-Man under “The Situation” Then students discuss the following questions: “How can you describe Ms. Pac-Man’s movements? What do you think “translation” means? What do you think “reflection” means? How can we get more precise to describe how far she translates, in what direction she rotates,...?” (8.G.2)
• Unit 3, Transformations and Angle Relationships, Lesson 3, Anchor Problem 3, “Below are the coordinate points from an original figure and a translated figure.  A(1,2) → A′(−1,−1)   B(−1,4) → B′(−3,1)  C(5,4) → C’(3,1) Describe the translation that occurred. Guiding Question: How can you answer this question without using a coordinate plane?”  (8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G.2, and 8.G.3)
• Unit 4, Functions, Lesson 2, Anchor Problem 2, students develop conceptual understanding of functions as they analyze two input/output tables and respond to the guiding questions, “Why is Table A not a function? How many different ways can you change Table A to make it a function? What needs to be true about Table B for it to not be a function?” (8.F.1)
• Unit 4, Functions, Lesson 5, Anchor Problem, Question 1, students are given four different graphs and answer: “Which one doesn’t belong? Why? What are some similarities that the graphs have?” (8.F.1) 
• Unit 4, Functions, Lesson 8, Anchor Problem 2, “Determine which equations below represent linear functions. Be prepared to justify your reasoning.” Equation 1: y = x2 + 1; Equation 2: y = 2x + 1; Equation 3: y = x/2;  Equation 4: $$y = x^3$$. Guiding Questions state: “What general conclusions can you make about equations of linear functions? About nonlinear functions?” (8.F.1, 8.F.3)
• Unit 6, Systems of Linear Equations, Lesson 10, Anchor Problem, Question 2 states, “The sum of two numbers is 361, and the difference between the two numbers is 173.” Students write a system of equations, and use any method to solve it. (8.EE.8.b)

Grade 8 materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. For example: 

• Unit 3, Transformations and Angle Relationships, Lesson 9, students "describe multiple rigid transformations using coordinate points." In the Target Task, students are given an image that has undergone two transformations, reflections across the x-axis and then translated 3 units left and 4 units up. “Explain how you can determine the coordinates for point E’ after the two transformations. Victoria determines that the new coordinates for point D after the two transformations will be (-5,5). She says that after the reflection, point D’ is located at (2-,1), and then the translation maps it to (-5,5). Is Victoria correct? Explain why or why not.” (8.G.1a, b, c and 8.G.3)
• Unit 4, Functions, Lesson 4 provides students the opportunity to independently demonstrate conceptual understanding of functions in the Problem Guidance Set, (EngageNY Mathematics Grade 8 Mathematics, Module 5, Topic A, Lesson 4, Problem Set) “You have just been served freshly made soup that is so hot that it cannot be eaten. You measure the temperature of the soup, and it is 210°F. Since 212°F is boiling, there is no way it can safely be eaten yet. One minute after receiving the soup, the temperature has dropped to 203°F. If you assume that the rate at which the soup cools is constant, write an equation that would describe the temperature of the soup over time.” (8.F.A)
• Unit 4, Functions, Lesson 8, students "determine if functions are linear or nonlinear when represented as table, graphs, and equations." In Target Task 2, students are given a table chart with some missing inputs and outputs, “Complete the table so it represents a linear function.” (8.F.1 and 8.F.3)
• Unit 4, Functions, Lesson 12, students "sketch graphs of functions given qualitative descriptions of the relationship." In the Target Task, “A child at a park sees a slide. She runs across the playground to the slide and slowly climbs up the stairs. At the top of the stairs, she pauses for a moment and then slides down to the bottom. Sketch two graphs for the situation: one that shows the child’s distance from the stairs of the slide as a function of time, and the other that shows the child’s speed as a function of time.” (8.F.5)
• Unit 5, Linear Relationships, Lesson 3, students "compare proportional relationships represented as graphs." In the Problem Set Guidance Activity, “Anna and Jason have summer jobs stuffing envelopes for two different companies. Anna earns $14 for every 400 envelops she finishes. Jason earns $9 for every 300 envelopes he finishes. Draw graphs and write equations that show the earnings, y as functions of the number of envelopes stuffed, n for Anna and Jason. Who makes more from stuffing the same number of envelopes? How can you tell this from the graph? Suppose Anna has savings of $100 at the beginning of the summer and she saves all her earnings from her job. Graph her savings as a function of the number of envelopes she stuffed. How does this graph compare to her previous earnings graph? What is the meaning of the slope in each case?” (8.EE.5)

The instructional materials for Match Fishtank Grade 8 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 Teacher 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 (8.EE.7, 8.EE.8b). For example:

• Unit 2, Solving One-Variable Equations, Lesson 1 develops procedural skills in solving linear equations. Anchor Problem 1, Number 2 states, “For each expression, write an equivalent simplified expression. Then verify that the expressions are equivalent by substituting a value in for x and solving: a. 2 + 3(x + 4);  b. 2 + 3(x -4);   c. 2 -3(x + 4);  d. 2 -3(x-4).”  (8.EE.7)
• Unit 5,  Linear Relationships, Lesson 6, students demonstrate procedural fluency when completing the Target Task to find slope. The Target Task states, “Find the slope of each graph below. Use a different pair of coordinate points in Graph B to show the slope of the line is the same through any two points on the line.” (8.EE.6)
• Unit 6, Systems of Linear Equations, Lesson 5, students "solve systems of linear equations using substitution when one equation is already solved for a variable." In Anchor Problem 2, “Solve the system below using substitution. Write your final answer as a coordinate point.”  2x−y=1 and y=14x+20  (8.EE.8.b)

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, Algebra by Example, and EngageNY, Great Minds. For example:

• Unit 2, Solving One Variable Equations, Lesson 3, students solve multi-step equations using the distributive property. Anchor Problem 3 states, “Solve the equation 2(3m + 6) - 4(1 - 2m) = -20.” Target Task, Problem 1 states, “Solve the equation below. For each step, explain why each line of your work is equivalent to the one before it.” (8.EE.7)
• Unit 2, Solving One Variable Equations, Lesson 5, Problem Set Guidance, provides students the opportunity to independently demonstrate procedural skills in simplifying expressions in problem 1 when students are asked to, “Simplify each expression: a. 5/3x - 4(x - 1/3)  b. -1/2(-8x + 6x + 10) - 2x” (8.EE.7)   
• Unit 6, Systems of Linear Equations, Lesson 5 provides students the opportunity to independently demonstrate procedural skills while using substitution to solve linear equations. For example, the problem states,  “Solve the system using substitution. Write your answer as a coordinate point. x+ 2y+ 4.”  (8.EE.8.b)
• Unit 6, Systems of Linear Equations, Lesson 8, students solve systems of equations using elimination. For example, Target Task Problem 2 states, “Solve the system. 9x + 2y = 9; 6x - 2y = -4.”  (8.EE.8.b)
• Unit 7, The Pythagorean Theorem and Volume , Lesson 14, students independently demonstrate procedural skill using formulas to find the volume of cones and cylinders. Target Task states, “Asher makes a cylinder with a radius of 3 inches and a height of 6 1/2 inches. How many cubic inches of clay did Asher use? Brandi makes a cone and uses approximately $$64in^3$$ of clay. The height of Brandi’s cone is 4 inches. What is the radius of the circular base of Brandi’s cone?” (8.G.9)

The instructional materials for Match Fishtank Grade 8 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 (8.EE.8.c, 8.F.B). 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 applications include but are not limited to:

• Unit 2, Solving One-Variable Equations, Lesson 4, students "Write and solve multi-step equations to represent situations, with variables on one side of the equation." For example, Anchor Problem 3, “Todd and Jason are brothers. Todd says, “I am twice as old as Jason was two years ago.” The sum of the brother’s ages is 38. How old is each brother?” (8.EE.7b)
• Unit 4, Functions, Lesson 9, Target Task, students compare properties of two functions where one is represented by an equation and the other by a graph: “Cora and Daniel are both saving money each month. The total amount in their savings accounts is a function of the number of months they have been saving. Who started with more money in their savings account? Who is saving at a faster rate? By how much? After 6 months, who will have more money in their savings account? ” (8.F.2)
• Unit 6, Systems of Linear Equations, Lesson 4, students "Solve real-world and mathematical problems by graphing systems of linear equations." For example, Anchor Problem 2, “Genny babysits for two different families. One family pays her $6 each hour and a bonus of $20 at the end of the night. The other family pays her $3 every half hour and a bonus of $25 at the end of the night. Write a system of equations that represents this situation and then graph the system. At what number of hours do the two families pay the same for babysitting services from Genny?” (8.EE.8.c)
• Unit 7, The Pythagorean Theorem and Volume, Lesson 11, students use the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. In Problem Set Guidance (Open Up Resources Grade 8 Unit 8 Practice Problems, Lesson 10, Problem 2)  “At a restaurant, a trash can’s opening is rectangular and measures 7 inches by 9 inches. The restaurant serves food on trays that measure 12 inches by 16 inches. Jada says it is impossible for the tray to accidentally fall through the trash can opening because the shortest side of the tray is longer than either edge of the opening. Do you agree or disagree with Jada’s explanation? Explain your reasoning.” (8.G.7)

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

• Unit 5, Linear Relationships, Lesson 12, Target Task, Problem 1 states, “Tickets to a concert are available for early access on a special website. The website charges a fixed fee for early access to the tickets, and the tickets to the concert all cost the same amount with no additional tax. A friend of yours purchases 4 tickets on the website for a total of $162. Another friend purchases 7 tickets on the website for $270. What function represents the total cost, y, for the purchase of x tickets on the website?” (8.EE.2.4, 8.F.2.4)
• Unit 6, Systems of Linear Equations, Lesson 1, Anchor Problem states, “Ivan’s furnace has quit working during the coldest part of the year, and he is eager to get it fixed. He decides to call some mechanics and furnace specialists to see what it might cost him to have the furnace fixed. Since he is unsure of the parts he needs, he decides to compare the costs based only on service fees and labor costs. Shown below are the price estimates for labor that were given to him by three different companies. Each company has given the same time estimate for fixing the furnace. Company A charges $35 per hour to its customers. Company B charges a $20 service fee for coming out to the house and then $25 per hour for each additional hour. Company C charges a $45 service fee for coming out to the house and then $20 per hour for each additional hour. For which time intervals should Ivan choose Company A, Company B, or Company C? Support your decision with sound reasoning and representations. Consider including equations, tables, and graphs.” (8.EE.8.C) 

The instructional materials in Grade 8 provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. For example:

• Unit 2, Solving One-Variable Equations, Lesson 7, students "Write and solve multi-step equations to represent situation, including variables on both sides of the equation." In the Target Task, “Melanie is looking for a summer job. After a few interviews, she ends up with two job offers. Blue Street Café pays $11.75 per hour plus $33 from tips each week. Fashion Icon Factory Store pays $14.50 per hour with no tips. If Melanie plans to work 10 hours per week, which job offer should she take to maximize her earnings? What if Melanie works 20 hours per week? How many hours would Melanie need to work in order for the pay at each job to be the same? Write and solve an equation.” (8.EE.7)
• Unit 4, Functions, Lesson 12, students "Sketch graphs of functions given qualitative descriptions of the relationship." In Anchor Problem 2, “Two stories are shown below. For each one, draw a graph to represent the functional relationship between the two quantities. Story 1: Distance from home vs. Time. You leave home and walk to the corner store. At the store, you spend a few minutes shopping, but then realize that you forgot your wallet at home. You run home, find your wallet immediately, and then run back to the store where you pay for your items. You leave the store and slowly walk away in the opposite direction of your home, toward the basketball court. Once you get to the court, you look for your friend for a few minutes, but when you realize that he’s not there, you walk quickly back home. Story 2: Temperature vs. Time. At 8 AM on a summer day, the temperature in Hartford, Connecticut, was 60°F. By 10 AM, the temperature had risen 10°F, where it stayed until 12 PM. From noon to 3 PM, the temperature rose to 85°F, after which it dropped at a steady rate until it hit 65°F at 9 PM." (8.F.5)
• Unit 5, Linear Relationships, Lesson 11, students "Write linear equations using slope and a given point on the line." In Anchor Problem 3, “A taxicab driver charges $2.40 per mile plus a one-time flat fee. A 3-mile ride costs you $10.30.  a. Write a function to represent the cost of a taxicab ride, y, for x miles.  b. How much will it cost you to travel 9 miles? (8.EE.2.6, 8.F.2.4)
• Unit 6, Systems of Linear Equations, Lesson 10, students "Solve real-world and mathematical problems using systems and any method of solution."  In Target Task, Problem 1, “Small boxes contain Blu-ray disks and large boxes contain one gaming machine. Three boxes of gaming machines and a box of Blu-rays weigh 48 pounds. Three boxes of gaming machines and five boxes of Blu-rays weigh 72 pounds. How much does each box weigh?” (8.EE.8.C)

The instructional materials for Match Fishtank Grade 8 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 8, Bivariate Data, Lesson 6, Target Task, students develop conceptual understanding while constructing scatter plots to interpret the relationship of the paired data. “According to the Bureau of Vital Statistics for the New York City Department of Health and Mental Hygiene, the life expectancy at birth (in years) for New York City babies is shown in the scatter plot below. An equation for a line fit to this data is represented by:  y=338x - 597.4. Explain what the slope of this equation model means in terms of the context. Use the model to predict the life expectancy for a baby born in New York City in 2020.” (8.SP.3)
• Unit 1, Exponents and Scientific Notation, Lesson 6, students, "Apply the power of powers rule and power of product rule to write equivalent, simplified exponential expressions." In Anchor Problem 3, “Lucas thinks that since $$(ab)^2 =a^2b^2$$, then that must mean $$(a+b)^2 =a^2+b^2$$. Is Lucas’ reasoning correct? Explain or show why or why not.”  (8.EE.1)
• Unit 8, Bivariate Data, Lesson 8, students "Calculate relative frequencies in two-way tables to investigate associations in data." In the Target Task,  “All the students at a middle school were asked to identify their favorite academic subject and whether they were in the 7th grade or 8th grade.” (A table is provided with the results.) “Is there an association between favorite academic subject and grade for students at the school? Support your answer by calculating appropriate relative frequencies using the given data.” (8.SP.4)

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 4, Lesson 12, Target Task, students develop their understanding of functions as they apply mathematics from verbal descriptions. “A child at a park sees a slide. She runs across the playground to the slide and slowly climbs up the stairs. At the top of the stairs, she pauses for a moment and then slides down to the bottom. Sketch two graphs for the situation: one that shows the child’s distance from the stairs of the slide as a function of time, and the other that shows the child’s speed as a function of time.” (8.F.1, 8.F.3, 8.F.5)
• Unit 5, Linear Relationships, Lesson 2, Anchor Problem 2, students engage in application and conceptual understanding as they graph and interpret the following situation, “Nia and Trey both had a sore throat so their mom told them to gargle with warm salt water. Nia mixed 1 teaspoon salt with 3 cups water. Trey mixed 1/2 teaspoon salt with 1 1/2cups of water. Nia tasted Trey’s salt water. She said, 'I added more salt so I expected that mine would be more salty, but they taste the same.’ Explain why the salt water mixtures taste the same. Find an equation that relates s, the number of teaspoons of salt, with w, the number of cups of water, for both of these mixtures. Draw the graph of your equation from part (b). Your graph in part (c) should be a line. Interpret the slope as a unit rate.” (8.EE.5) 
• Unit 6, Systems of Linear Equations, Lesson 11, in Target Task, Problem 2, students engage in procedural skill and application. For example, Problem 2 states, “Kim has a small container and a large container as shown. It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container. What is the size of each of the two containers?” (8.EE.8.c)

The instructional materials reviewed for Match Fishtank Grade 8 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 MPs are addressed and what mathematically proficient students should do. For Example, Unit 4: Functions, Unit Summary states, “As students progress through the unit, they analyze functions to better understand features such as rates of change, initial values, and intervals of increase or decrease, which in turn enables students to make comparisons across functions even when they are not represented in the same format. Students analyze real-world situations for rates of change and initial values and use these features to construct equations to model the function relationships (MP.4).” 
• Lessons usually include indications of Mathematical 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 2 Solving One Variable Equations, Lesson 5, Criteria For Success states, “Model a situation using an equation and make adjustments to the model as the situation changes (MP.4). De-contextualize a situation to represent it algebraically, and re-contextualize to interpret the solution in context of the problem (MP.2).” Lesson 8, Anchor Problem 2 states, “Students make use of structure to decide how to categorize each equation, looking for equations that have the same variable term on both sides of the equation and considering the values of any constants (MP.7). Lesson 9, Tips for Teachers states, “In this lesson, students use what they know about equations with no, infinite, or unique solutions to reason about solutions without having to completely solve the equation. For example, if they can simplify each side of an equation to one or two terms, they should be able to determine if a solution is possible and if so, if that solution is unique (MP.7).”
• In some Problem Set Guidances, MPs are identified within the problem. For example, Unit 4 Functions, Lesson 11, MARS Formative Assessment Lesson for Grade 8 Interpreting Distance-Time Graphs states, “This lesson also relates to all the Standards for Mathematical Practice, with a particular emphasis on: 2. Reason abstractly and quantitatively, 7. Look for and make use of structure and 8. Look for and express regularity in repeated reasoning.”

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

• MP7 enriches the mathematics in Unit 5, Lesson 6, as students analyze and find slope from a graph. Anchor Problem 2, Notes, states, “Before finding the slope, ask students to take a close look at the lines. Is the line rising to the left or right? Does the line appear steep or shallow? Using these observations of the lines, students can better their understanding of the numerical value of the slope as positive or negative, and the absolute value as greater than 1 or less than 1 (MP.7).”
• MP2 enriches the mathematical content in Unit 2, Lesson 4, Criteria for Success, as students “decontextualize a situation to represent it algebraically, and re-contextualize to interpret the solution in context of the problem”. Anchor Problem 3 states, “Todd and Jason are brothers. Todd says, 'I am twice as old as Jason was two years ago.' The sum of the brothers' ages is 38. How old is each brother?” 
• MP8 enriches the mathematical content in Unit 5, Lesson 6, Anchor Problem 3, Notes, states that when students use “understanding of similar triangles to reach the conclusion that the slope of a line is the same through any two points on the line”. “Using any two points on the line, you can create similar triangles, which will have side lengths in the same ratio. Since slope is the ratio of the vertical height of the triangle to the horizontal width, you can conclude that the slope through any two points will be the same.” Throughout Problem Set Guidance, students are provided opportunities to use two points on a line to find the slope. Target Task, Question 2 states, “Use a different pair of coordinate points in Graph B to show the slope of the line is the same through any two points on the line.”
• MP4 enriches the mathematics in Unit 5, Linear Relationships, Lesson 4, Anchor Problem 1, Notes states, “Students can approach this problem in several ways using their understanding of proportional relationships (MP.4). For example, they may graph a line representing the new machine in the graph alongside the old machine. They may write an equation to represent each machine. They may determine the rate of change for each machine and use a proportion.” “At a factory, a machine fills jars with salsa. The manager of the factory is considering buying a new machine that will fill 78 jars of salsa every 3 minutes. To support his decision, he wants to compare the rate of the new machine to the rate of the old machine that is currently in the factory. The graph below shows the number of jars of salsa filled over time with the old machine.The manager is about to fill an order of 765 jars of salsa. How long would it take to fill this order on each machine? Should the manager consider replacing the old machine with the new one? Explain.” There is no evidence where MPs are addressed separately from the grade-level content.

The instructional materials reviewed for Match Fishtank Grade 8 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 Problems 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 in solving them.

• Unit 6, Systems of Linear Equations, Lesson 10, Anchor Problem 3, students “make sense of how this information fits together, what unknowns they are looking to solve for, and how they can represent the relationships with equations that can be solved.” For example, Anchor Problem states, “A type of pasta is made of a blend of quinoa and corn. The pasta company is not disclosing the percentage of each ingredient in the blend, but we know that the quinoa in the blend contains 16.2% protein and the corn in the blend contains 3.5% protein. Overall, each 57-gram serving of pasta contains 4 grams of protein. How much quinoa and how much corn is in one 57-gram serving of the pasta?”
• Unit 7, Pythagorean Theorem and Volume, Lesson 16, Anchor Problem 3 states, “Option #2 (Students choose or are assigned one of the five Anchor Problems to work on with a small group of peers. After collaboratively developing a strategy and preparing a solution, students should create a poster of their response.)” Shipping Rolled Oats: “Rolled oats (dry oatmeal) come in cylindrical containers with a diameter of 5 inches and a 9 1/2 height of inches. These containers are shipped to grocery stores in boxes. Each shipping box contains six rolled oats containers. The shipping company is trying to figure out the dimensions of the box for shipping the rolled oats containers that will use the least amount of cardboard. They are only considering boxes that are rectangular prisms so that they are easy to stack. a. What is the surface area of the box needed to ship these containers to the grocery store that uses the least amount of cardboard? b. What is the volume of this box?” In the Criteria for Success, students “Map out a solution pathway and use relevant formulas and math concepts to solve complex, real-world problems. (MP.1 and MP.4)” 

MP.2: Students reason abstractly and quantitatively.

• Unit 2, Solving One Variable Equations, Lesson 4, students “decontextualize a situation to represent it algebraically, and re-contextualize to interpret the solution in the context of the problem.” For example, Anchor Problem 1 states, “The length of a rectangle is 3 cm less than twice the width of the rectangle. If the perimeter is 75 cm, what are the dimensions of the rectangle?”
• Unit 8, Bivariate Data, Lesson 1, Anchor Problem 1, students “interpret ordered pairs (x,y) in scatter plots in context of the variables.” For example, students are shown a graph of weight and price for sugar and asked, “Each point on this graph represents a bag of sugar. Which point shows the heaviest bag? Which point shows the cheapest bag? Which points show bags with the same weight? Which points show bags with the same price? Which of F or C gives the best value for the money? How can you tell?”

MP.4: Students model with mathematics.

• Unit 4, Functions, Lesson 10, in Anchor Problem 1, students model a real-world context by comparing the properties of two functions represented in different ways. For example, “Sam wants to take his music player and his video game player on a car trip. An hour before they plan to leave, he realized that he forgot to charge the batteries last night. At that point, he plugged in both devices so they can charge as long as possible before they leave. Sam knows that his music player has 40% of its battery life left and that the battery charges by an additional 12 percentage points every 15 minutes. His video game player is new, so Sam doesn’t know how fast it is charging but he recorded the battery charge for the first 30 minutes after he plugged it in. If Sam’s family leaves as planned, what percent of the battery will be charged for each of the two devices when they leave? How much time would Sam need to charge the battery 100% on both devices? Students may choose to use any representation (equation, table, graph) for these functions in order to analyze and compare them.” 
• Unit 7, Pythagorean Theorem and Volume, Lesson 11, Anchor Problem 1 states, “Act 1: Watch the Taco Cart by Dan Meyer. What do you notice? What do you wonder? Who will reach the taco cart first? Make a guess. Act 2: Ask students what information they need to determine who will reach the taco cart first. Share the information below as it is requested. Note, the speeds refer to the speed of walking on the sand and the speed of walking on the concrete sidewalk. Act 3: Once students have reached their solutions, share the video answer.” Anchor Problem Notes state, “students model a real-world situation using the Pythagorean Theorem. They first make a guess and predict who will reach the taco cart first. Then, as new information is received, students determine how they will use that information to design and modify a model to accurately answer the question (MP.4).”
• Unit 8, Bivariate Data, Lesson 5, Anchor Problem 2, students model and analyze data related to the relationship between time and percent battery charge. “Jerry forgot to plug in his laptop before he went to bed. He wants to take the laptop to his friend’s house with a full battery. The pictures below show screenshots of the battery charge indicator after he plugs in the computer at 9:11 A.M. The screenshots suggest an association between two variables. What are the two variables in this situation? Make a scatter plot of the data. Draw a line that fits the data and find the equation of the line. When can Jerry expect to have a fully charged battery?” Anchor Problem Notes state, “This problem engages students in MP.4, and asks students to analyze the relationship between time and percent battery charge. Information is given in an unconventional way and students must determine what the variables are, how to represent the relationship between then visually, and then what conclusions they can make.”

MP.5: Students use appropriate tools strategically.

• Unit 3, Transformations and Angle Relationships, Lesson 1, Anchor Problem 2 states, “For each pair of figures, decide whether these figures are the same size and same shape. Be prepared to justify your reasoning. You may use mathematical tools to make your decision.” Lessons in this unit ensure the full depth of MP5 by emphasizing student choice. 
• Unit 4, Functions, Lesson 8, Anchor Problem 2 states, “Determine which equations below represent linear functions. Be prepared to justify your reasoning; equation 1:  $$y = x^2 +1$$, equation 2:  y = 2x +1, equation 3:  $$y = x^2$$, equation 4:  $$y = x^2$$."  “This should be an investigatory problem for students to explore using different strategies and tools.”

MP.6: Students attend to precision.

• Unit 1, Exponents and Scientific Notation, Lesson 2, Anchor Problem 3 states, “Students attend to precision as they use parentheses appropriately when substituting in negative values in order to convey multiplication and not subtraction. Evaluate the expression when x = 2 and y = -3; $$xy^2 + xy$$.” 
• Unit 3, Transformations and Angle Relationships, Lesson 7, Anchor Problem 1 states, “Figure 1 is shown in the coordinate plane below. Which figure(s) would Figure 1 map to if it were a. translated? b. reflected? c. rotated? The vertices are intentionally not named in this problem. Ensure students to use precision in their communication, adding labels or names as needed.”                 

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

• Unit 2, Solving One Variable Equations, Lesson 8, Anchor Problem 2, students look for patterns and structures in one variable equations. For example, the Anchor Problem 2 states, “Sort the equations below into the three categories Guiding Questions:  a. Ask yourself for each problem, is there a value that would make the equation true? If so, is there more than one value for ?  b. At first look, how would you classify  3x = 2x?  If you were to solve it, how would you go about it and what would the solution be?  c. What other strategies can you use to categorize these equations?”
• Unit 7, Pythagorean Theorem and Volume, Lesson 9, in Anchor Problem 2, students demonstrate an understanding of structure as they explain how to use the Pythagorean Theorem on triangles that are not right triangles. For example, students are told to “Find the area of the isosceles triangle below. Give your answer to the nearest tenth of a unit. This Anchor Problem highlights how knowing the Pythagorean Theorem can offer additional insight into other triangles and shapes by identifying the opportunity to create right triangles.” 
• Unit 7, Pythagorean Theorem and Volume, Lesson 16, Anchor Problem 2, students use their understanding of the overall structure of the formulas to solve for surface area of a cone and surface area of a sphere. For example, students are told"The diagram shows three glasses (not drawn to scale). The measurements are all in centimeters. The bowl of glass 1 is cylindrical. The inside is 5 cm and the inside height is 6 cm. The bowl of glass 2 is composed of a hemisphere attached to a cylinder. The inside diameter of both the hemisphere and the cylinder is 6 cm. The height of the cylinder is 3 cm. The bowl of glass 3 is an inverted cone. The inside diameter is 6 cm and the inside slant height is 6 cm. a. Find the vertical height of the bowl of glass 3. b. Calculate the volume of the bowl of each of these glasses. c.  Glass 2 is filled with water and then half the water is poured out. Find the height of the water.”

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

• Unit 3, Transformations and Angle Relationships, Lesson 5, Anchor Problem 1, students explain patterns, discuss methods and solution strategies, and evaluate the result of a reflection over x and y axis. For example, the Anchor Problem 1 states, “Triangle LMN underwent a single transformation to become triangle PQR, shown below. a. What single transformation maps triangle LMN to triangle PQR ? Describe in detail. b. Name two things that are the same about both triangles. c. Name two things that are different about the triangles.”
• Unit 7, Pythagorean Theorem and Volume, Lesson 4, in Anchor Problem 2,  students “Follow the directions to approximate the location of 2/11 on the number lines below. a. On the topmost number line, label the tick marks. Next, find the first decimal place of 2/11 using long division and estimate where should be placed on the top number line. b. Label the tick marks of the second number line. Find the next decimal place of 2/11 by continuing the long division and estimate where should be placed on the second number line. Add arrows from the second to the third number line to zoom in on the location of 2/11. c. Repeat the earlier step for the remaining number lines. d) What do you think the decimal expansion of 2/11 is?” In this problem, students investigate the structure of the fraction 2/11, as it results in a repeated pattern in the decimal equivalent.”

The instructional materials reviewed for Match Fishtank Grade 8 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, Solving One Variable Equations, Lesson 3, Anchor Problem 2 states, “Two more students, Christian and Esther, are solving the same equation. They take a different approach to solving the equation, but they each make an error in the first two lines of their work, shown below. 15 - 3(x - 2) + 6x = 3(13). Explain the error that each student made.”
• Unit 7,  Pythagorean Theorem and Volume, Lesson 1, Target Task states, “The square below has an area of 20 square units. Taylor writes the equation $$s^2=20$$  to find the measure of the side length of the square. She reasons that the solution to the equation is $$\sqrt20$$ and concludes that the side length of the square is 10 units. Do you agree with Taylor? Explain why or why not. If you disagree, include the correct side length of the square.”
• Unit 7, Pythagorean Theorem and Volume, Lesson 2, Problem Set Guidance,  (EngageNY Mathematics, Grade 8 Mathematics, Module 7, Topic B, Lesson 11, Problem Set) Question 8 states, “Henri computed the first 100 decimal digits of the number 352/541 and got $$0.6506469500924214417744916820702402957486136783733826247689463955677079482439926062846580406654343807763401109057301294$$…. He saw no repeating pattern to the decimal and so concluded that the number is irrational. Do you agree with Henri’s conclusion? If not, what would you say to Henri?”
• Unit 8, Bivariate Data, Lesson 6, Problem Set Guidance, (EngageNY Mathematics, Grade 8 Mathematics, Module 6, Topic C, Lesson 10), Problem Set, Problem 3 states, “Simple interest is money that is paid on a loan. Simple interest is calculated by taking the amount of the loan and multiplying it by the rate of interest per year and the number of years the loan is outstanding. For college, Jodie’s older brother has taken out a student loan for $4,500 at an annual interest rate of 5.6%, or 0.056. When he graduates in four years, he has to pay back the loan amount plus interest for four years. Jodie is curious as to how much her brother has to pay.  a. Jodie claims that her brother has to pay a total of $5,508. Do you agree? Explain. As an example, a $1,200 loan has an 8% annual interest rate. The simple interest for one year is $96 because (0.08)(1200) = 96. The interest for two years would be $192 because (2)(96) = 192.  b. Write an equation for the total cost to repay a loan of $???? if the rate of interest for a year is ???? (expressed as a decimal) for a time span of ???? years.  c. If ???? and ???? are known, is the equation a linear equation?  d. In the context of the problem, interpret the slope of the equation in words.  e. In the context of the problem, interpret the ????-intercept of the equation in words. Does interpreting the intercept make sense? Explain.”

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

• Unit 1, Exponents and Scientific Notation, Lesson 8, Target Task states, “What errors were made in the examples below? Explain the mistake and then find an equivalent expression. $$x^3y^2/y^4x^2 = -1/y^6x^5$$”
• Unit 4, Functions, Lesson 2, Target Task states, “A certain business keeps a database of information about its customers. Let C be the rule that assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning. Let P be the rule that assigns to each phone number in the table above, the customer name associated with it. Is P a function? Explain your reasoning. Explain why a business would want to use a person's social security number as a way to identify a particular customer instead of their phone number.”
• Unit 5, Linear Relationships, Lesson 4 states, “Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. The graph below represents the rate at which Pipe B can fill the same-sized buckets. Write a linear equation that represents the number of buckets, y, that Pipe A can fill in x minutes. Which pipe fills buckets faster? Justify your answer.”
• Unit 6, Systems of Linear Equations, Lesson 11, Problem Set Guidance MARS Summative Assessment Tasks for Middle School Chips and Candy states,   "4. Clancy has just $1. Does he have enough money to buy a bag of potato chips and a candy bar? Explain your answer by showing your calculation.”
• Unit 7,  Pythagorean Theorem and Volume, Lesson 4, Problem Set Guidance, (Open Up Resources, Grade 8 Unit 8 Practice Problems) Lesson 14, Question 1 states, “Andre and Jada are discussing how to write 17/20 as a decimal. Andre says he can use long division to divide 17 by 20 to get the decimal. Jada says she can write an equivalent fraction with a denominator of 100 by multiplying 5/5, then writing the number of hundredths as a decimal. Do both of these strategies work? Which strategy do you prefer? Explain your reasoning.”
• Unit 8, Bivariate Data, Lesson 3, Target Task Problem 1 states, “Which of the following scatter plots shows a negative linear relationship? Explain how you know.”

The instructional materials reviewed for  Math FishTank Grade 8 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, Exponents and Scientific Notation, Lesson 4, Anchor Problem 2. Teachers are prompted to have students review incorrect statements. Guiding Questions are as follows: a. “What mistake was made in the first example?” b. “Before correcting the problem, what are other possible answers – right and wrong – that might represent the product? (For example, $$8^8$$, $$16^8$$, $$16^15$$, $$64^8$$, $$64^15$$ etc.)” c. “What strategy can you use to find the correct answer without evaluating the exponent? Can you use the same strategy for the second example? How convincing is your reasoning? How do you know your new answer is correct?” 
• Unit 1, Exponents and Scientific Notation, Lesson 14, Guiding Questions are provided for teachers to support students in constructing viable arguments and analyzing the errors of others: a. “When multiplying with numbers in scientific notation, you are able to use the commutative property to multiply the two first factors together and then the two powers of 10. Why does this not work with addition?” b)\. “What is the place value of the “2” in 2.5×103?” c. “What is the place value of the “1” in $$1.3×10^4$$?” d.  “Can you add these two numbers together to get the digit ‘3’? Why or why not?” e. What do you notice about the powers of 10 in the two numbers? Why is this an important detail when adding numbers?”
• Unit 2,  Solving One Variable Equations, Lesson 8, Anchor Problem 2 states, “Sort the equations below into the three categories. Ask yourself for each problem, is there a value for x that would make the equation true? If so, is there more than one value for x? At first look, how would you classify 3x = 2x? If you were to solve it, how would you go about it and what would the solution be? What other strategies can you use to categorize these equations? Once students have sorted the equations into the three categories, have them defend their decisions by explaining the similarities between each equation within each category.” 
• Unit 5, Linear Relationships, Lesson 6, Anchor Problem 1 states, “Three staircases are shown below. Which staircase is the steepest? Without doing any calculations, order the staircases from least steep to most steep. Slope is a measure of steepness. The greater the slope, the greater the steepness. Use the measurements provided in the diagram to justify the order of steepness you determined earlier.  Another staircase climbs 8 feet over a distance of 10 feet. Where does this staircase fall in the order of steepness?” Guiding Questions are as follows:  a. “How did you determine your order without doing any calculations?” b. “Discuss your order with a peer. Do you have the same order? How did you each determine your order? Will you change your order based on your conversation?” c. “What measurements are important to consider when finding slope? How do you use those measurements to determine slope?” d. “Explain how to find the steepness of any staircase that climbs a distance of y feet over a distance of x feet. How does this relate to the concept of slope? “Notes include the following: “Students will likely be able to determine the order of steepness intuitively by looking at the staircases. They are then challenged to find the numerical representation that supports their order. Some students may consider the area of the triangles formed by the staircase, or the fraction of width over height, or just consider the height in isolation; however, none of these measurements match the order of steepness A, C, B from least to greatest. Discuss the various strategies that come up in class and which seem valid and why”.

The instructional materials reviewed for Match Fishtank Grade 8 meet the expectations that materials use accurate mathematical terminology.

The Match Fishtank Grade 8 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 3, Transformations and Angle Relationships’ vocabulary includes the following words: translation, reflection, rotation, rigid transformation, dilation, congruent/congruence, scale factor, corresponding angles, alternate interior and exterior angles, vertical angles, and similar.  
• Unit 4, Functions, Lesson 1, Criteria For Success states, “Define a function as a relationship between two sets in which every input has exactly one output.”

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 3, Transformations and Angle Relationships, Lesson 4, Anchor Problem 1 states, “Use this problem to ensure students know a reflection is described by a line of reflection from which every point on the original image is the same distance as each corresponding point on the reflected image.”
• Unit 7, Pythagorean Theorem and Volume, Lesson 3, Anchor Problem 2 states, “When estimating the values, students must determine how precise they need to be in their estimates in order to find an appropriate location on the number line. For example, students may determine that $$\sqrt50+1$$ is between 8 and 9, but they must then determine if they should plot the point closer to the 8, closer to the 9, in the middle, etc.”
• Unit 8, Bivariate Data, Lesson 3, Anchor Problem 1. “In this Anchor Problem, students make the connection between positive linear associations and lines with positive slopes, and negative linear associations and lines with negative slopes. The language and vocabulary is scaffolded for students, as it is given to them in part (d). Throughout the rest of the lesson, listen for students using this language in other examples, and if need be, direct them back to this problem for reminders or guidance.”  

The Match Fishtank Grade 8 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, Functions, Lesson 7, Anchor Problem 1 states, “What do you think a linear function is? What does it look like?” 
• Unit 5, Linear Relationships, Lesson 6, Anchor Problem 1 states, “What does slope mean? How do you measure it? These graphs do not include measurements like the staircases. How can you determine the measurements you need to find the slope?”
• Unit 7, Pythagorean Theorem and Volume, Lesson 5, Anchor Problem 1 states, “Recall the definition of a rational number. Why are all of these rational numbers?”