6th Grade - Gateway 1

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into eight units and each unit contains a Pre-Unit Assessment, Mid-Unit Assessment, and Post-Unit Assessment. Pre-Unit assessments may be used “before the start of a unit, either as part of class or for homework.” Mid-Unit assessments are “designed to assess students on content covered in approximately the first half of the unit” and may also be used as homework. Post-Unit assessments “are designed to assess students’ full range of understanding of content covered throughout the whole unit.” Examples of Post-Unit Assessments include: 

• In Unit 1, Post-Unit Assessment, Understanding and Representing Ratios, Problem 2 states, “All of the benches in a park are red or blue. The ratio of red benches to blue benches in the park is 3:4. Based on this information, which of the following statements is true? a. For every 4 benches in the park, 3 are red. b. For every 7 benches in the park, 4 are red. c. For every 3 red benches in the park, there are 4 blue benches. d. For every 3 red benches in the park, there are 7 blue benches.” (6.RP.1)

• In Unit 4, Post-Unit Assessment, Rational Numbers, Problem 5 states, “Ethan is hiking in a canyon. He is at an elevation that is below sea level. His elevation is within 200 feet of sea level. Which of the following could be Ethan’s elevation in feet? a. -300, b. -150, c. 150, d. 300.” (6.NS.5)

• In Unit 5, Post-Unit Assessment, Numerical and Algebraic Expressions, Problem 5 states, “What is the value of the expression below? 2[3(4² + 1)] − 2³. a. 156  b. 110 c. 94 d. 48.” (6.EE.1) 

• In Unit 8, Post-Unit Assessment, Statistics, Problem 1 states, “Which questions are statistical questions? Select two correct answers. a. How old is Mr. Patterson? b. How many states has Juanita visited? c. How many students are in Mrs. Lee’s class today? d. How many students eat lunch in the cafeteria each day? e. How many pets does each student at your school have at home?” (6.SP.1)

The instructional materials reviewed for Fishtank Plus Math Grade 6 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The instructional materials provide extensive work in Grade 6 by including Anchor Problems, Problem Sets, and Target Tasks for all students in each lesson. Within Grade 6, students engage with all CCSS standards. Examples of problems include:

• In Unit 1, Understanding and Representing Ratios, Lesson 10 engages students with extensive work for 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems). In Anchor Problem 3, students complete a ratio table and then reason about the relationships between numbers. It states, “A cleaning solution can be made by mixing bleach and water. The ratio table below has some missing information. Fill in the blanks.” Guiding Questions include, “What is the ratio of tablespoons of bleach to gallons of water used in the cleaning solution? What is the multiplicative relationship between 3 tablespoons of bleach and 6 tablespoons of bleach? How can you use this to determine the number of gallons of water to use with 6 tablespoons of bleach? Why would 7 gallons of water for 6 tablespoons of bleach be incorrect? What mistake was made? Once you determine the ratio of 9:, how can you use that to determine the ratio of 90:__?” 

• In Unit 5, Numerical and Algebraic Expressions, Lesson 2 engages students with extensive work with grade-level problems for 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents). In Anchor Problem 3, students evaluate numerical expressions that involve whole-number exponents. It states, “Evaluate each numerical expression. a. 2×(3+4^2) b. 2×(3+4)^2 c. [2(3+4)]^2 d. 2+3(4+1)^2.”

• In Unit 8, Statistics, Lesson 10 engages students with extensive work with grade-level problems for 6.SP.3 (Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number). In Problem Set, Problem 6, students describe data sets by comparing measures of center and measures of spread. It states, “Cora is preparing to go out for the day. She wants to make sure she has the right clothing with her. Which information about the day’s temperature would be more helpful for her to know: the range of predicted temperatures or the average of predicted temperatures? Explain your reasoning.”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 6 standards through a consistent lesson structure, including Anchor problems, Problems Sets, and Target Tasks. Anchor Problems include a connection to prior knowledge, multiple entry points to new learning, and guided instruction support. Problem Set Problems engage all students in practice that connects to the objective of each lesson. Target Task Problems can be used as formative assessment. Examples of meeting the full intent include:

• In Unit 2, Unit Rates and Percents, Lesson 5 provides the opportunity for students to engage with the full intent of standard 6.RP.3b (Solve unit rate problems including those involving unit pricing and constant speed). In the Target Task, students calculate unit rates to determine which is the better buy. It states, “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?” In the Problem Set, Problem 4, students engage with the remaining part of the standard by finding constant speed. It states, “Jada bikes 2 miles in 12 minutes. Jada’s cousin swims 1 mile in 24 minutes. A. Who was moving faster? How much faster? B. One day Jada and her cousin line up on the end of a swimming pier on the edge of a lake. At the same time, they start swimming and biking in opposite directions. i. How far apart will they be after 15 minutes? ii. How long will it take them to be 5 miles apart?”

• In Unit 4, Rational Numbers, Lessons 2 and 3, engage students with the full intent of 6.NS.5 (Understand that positive and negative numbers are used together to describe quantities having opposite directions or values and use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation). In Lesson 2, Problem Set, Problem 6, students use and understand positive and negative numbers to represent real-world context. It states, “For each description below, write an integer to represent the situation. Then represent the integer on a number line, indicating the location of 0 and using an appropriate scale. A. You earn $35 for babysitting. B. You owe your sister $18. C. The average temperature in Boston last week was 6 degrees Celsius below 0.” In Lesson 3, Problem Set, Problem 2, students explain the meaning of 0 in real-world contexts. It states, “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. 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.”

• In Unit 6, Equations and Inequalities, Lessons 4, 6, and 7, engage students with the full intent of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers). In Lesson 4, Anchor Problem 2, students reason about and solve one step equations. It states, “Draw a tape diagram to represent the equation d - 5 = 7; then use it to find the value of d. How can you solve the equation p - \frac{3}{4} = 8 without using a diagram?” In Lesson 6, Target Task, students use an equation to solve a real world problem involving percent. It states, “Paula is saving money to buy a tablet. So far, she has saved $54, which is 45 percent of what she needs to buy the tablet. Write and solve an equation to find the price of the tablet.” In Lesson 7, Problem Set, Problem 2, students write and solve an equation in the form of px = q for a real-world problem. It states, “Clint is trying to figure out how many different pizza slices were sold last week at the local pizza shop. He knows that the total number of slices sold was 680. He also knows that: Twice as many slices of cheese were sold as the number of two-topping slices were sold. Half as many two-topping slices were sold as one-topping slices were sold. The highest number of toppings anyone can get on a slice is 2. How many slices of each number of toppings were sold last week? Draw a tape diagram to represent the slices, then write and solve an equation.”

While materials meet standards for this indicator, students do not have the opportunity for extensive work with 6.NS.8 (Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate). For example, in Unit 4, Lesson 13, Anchor Problem 1 is a real world graphing problem but students do not graph in all four quadrants. Other real world problems for this standard are not included.

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade: 

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 5.5 out of 8, approximately 69%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 83 out of 126, approximately 66%. The total number of lessons include: 118 lessons plus 8 assessments for a total of 126 lessons. 

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work) is 98 out of 143, approximately 69%. There are a total of 17 flex days and 15 of those days are included within units focused on major work. By adding 15 flex days focused on major work to the 83 lessons devoted to major work, there is a total of 98 days devoted to major work.

• The number of days devoted to major work (excluding flex days, while including assessments and supporting work connected to the major work) is 83 out of 126, approximately 66%. While it is recommended that flex days be used to support major work of the grade within the program, there is no specific guidance for the use of these days.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 66% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. These connections are sometimes listed for teachers as “Foundational Standards'' on the lesson page. Examples of connections include:

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 12, Target Task connects the supporting work of 6.NS.3 (fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.RP.A (understand ratio concepts and use ratio reasoning to solve problems). “Students divide decimals by using the standard algorithm” in the Target Task. It states, “Sophia’s dad paid $43.25 for 12.5 gallons of gas. a. What is the cost of one gallon of gas? b. Approximately how many gallons of gas can you get for $1? Round to the nearest hundredth.”

• In Unit 5, Numerical and Algebraic Expressions, Lesson 9, Problem Set, Problem 7 connects the supporting work of 6.NS.4 (find the greatest common factor of two whole numbers and the least common multiple of two numbers and use the distributive property to express a sum of two whole numbers) to the major work of 6.EE.3 (apply the properties of operations to generate equivalent expressions). “Students write equivalent expressions using the distributive property” in Problem 7. It states, “The expressions below are shown in expanded form.  Write each expression in factored form. A. 72x + 60 B. 124y + 64 C. 35x + 119 D. 16h + 192w.” 

• In Unit 7, Geometry, Lesson 7, Anchor Problem 1 connects the supporting work of 6.G.3 (draw polygons in the coordinate plane given coordinates for the vertices) to the major work of 6.NS.6 (understand a rational number as a point on the number line). In the Anchor Problem, students use a coordinate plane to graph polygons and find the area and perimeter. It states, “A rectangle has three vertices at points A(4,5), B(4,−3), C(−5,−3). a. What are the coordinates of the fourth vertex, D, of the rectangle? If needed, use the coordinate plane below. b. What is the area of the rectangle? c. What is the perimeter of the rectangle?”

• In Unit 8, Statistics, Lesson 13, Target Task connects the supporting work of 6.SP.4 (display numerical data in plots on a number line, including dot plots, histograms, and box plots) to the major work of 6.RP.3c (find a percent of a quantity as a rate per 100). In the Target Task, students analyze a circle graph in order to determine missing fractional or percentage amounts in order to solve the real world problem. It states, “The circle graph below shows Jordan’s weekly exercise. Jordan spends an equal amount of time playing basketball and doing dance each week. If she spent $$1\frac{1}{2}$$ hours playing basketball in a week, then how many total hours did Jordan spend exercising during the week?”

The instructional materials for Fishtank Plus Math Grade 6 meet expectations that materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Materials are coherent and consistent with the Standards. Examples of connections include:

• In Unit 5, Numerical and Algebraic Expressions, Lesson 3 connects the major work of 6.EE.A (apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.EE.B (reason about and solve one-variable equations and inequalities). In Problem Set, Problem 2, students use variables to write algebraic expressions. It states, “A packing company has 2 boxes. Each box has a width of 20 inches and a length of 30 inches, but they have different volumes. The formula for the volume of a box is l × w × h = v  A. If the height of the first box is 12 inches, what is the volume of the box? B. If the height of the second box is 8 inches, what is the volume of the box? C. A third box has the same width and length as the first two. Write an expression that represents the volume of the third box.” 

• In Unit 6, Equations and Inequalities, Lesson 13 connects the major work of 6.EE.C (represent and analyze quantitative relationships between dependent and independent variables) to the major work of 6.RP.A (understand ratio concepts and use ratio reasoning to solve problems). In Problem Set, Problem 3, students represent the relationship between two quantities in graphs, equations, and tables. It states, “The graph represents the amount of time in hours it takes a ship to travel various distances in miles.” A graph is shown with the x-axis labeled “distance traveled (miles)” with intervals of 25 up to 250, the y-axis labeled “time (hours)” with intervals of 2 up to 10. There are 10 points plotted on the graph including (25,1) (50,2) (75,3) (100,4). Students complete the following: “a. Write the coordinates of one of point on the graph. What does the point represent? b. What is the speed of the ship in miles per hour? c. Write an equation that relates the time, t, it takes to travel a given distance, d.”

• In Unit 7, Geometry, Lesson 9 connects supporting work of 6.G.A (solve real-world and mathematical problems involving area, surface area, and volume) to the supporting work of 6.NS.B (compute fluently with multi-digit numbers and find common factors and multiples). In Problem Set, Problem 1, students use their understanding of integers to represent and find the area of polygons on the coordinate plane. It states, “Brad’s neighborhood is defined by a pentagonal region on a map given by the coordinate points below. A:(−6, 5) B:(5, 5) C:(7, 0) D:(5, −3) E:(−6, −3). Draw the region that defines Brad’s neighborhood and determine the area of the neighborhood in square miles. Each square on the grid represents 0.01 square miles.” A coordinate grid is provided, with x-axis values from -10 to 10 and y-axis values of -10 to 10.

• In Unit 8, Statistics, Lesson 5, Target Task connects the supporting work of 6.SP.B (summarize and describe distributions) to the supporting work of 6.NS.B (compute fluently with multi-digit numbers and find common factors and multiples). In the Target Task, students read a dot plot and use division to calculate an average. It states, “What is the average number of miles that this group of sixth graders live from school?” A dot plot titled, “Number of miles from school” includes values from 1 to 9.

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

Content from future grades is identified within materials and related to grade-level work. These references are consistently included within the Unit Summary. Examples include:

• In Unit 1, Understanding and Representing Ratios, Unit Summary connects 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the work of seventh grade, eighth grade, and high school. Unit Summary states, “Students extend their understanding of ratios and rates to investigate proportional relationships in seventh grade. This sets the groundwork for the study of functions, linear equations, and systems of equations, which students will study in eighth grade and high school.” (7.RP.A, 8.EE.B, 8.EE.C)

• In Unit 2, Unit Rates and Percent, Unit Summary connects 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the work of seventh grade. Unit Summary states, “In seventh grade, students will solve even more complex ratio, rate, and percent problems involving, for example, tax and percent increase or decrease. Seventh graders will investigate and analyze proportional relationships between quantities, and use more efficient and abstract methods to solve problems.” (7.RP.2, 7.RP.3)

• In Unit 4, Rational Numbers, Unit Summary connects 6.NS.C (Apply and extend previous understanding of numbers to the system of rational numbers) to the work of seventh grade. Unit Summary states, “In seventh grade, students will discover how to compute with rational numbers and what happens when the properties of operations are applied to negative values. The work they do in this sixth-grade unit is foundational of these seventh-grade concepts.” (7.NS.1, 7.NS.2, 7.NS.3)

• In Unit 7, Geometry, Unit Summary connects 6.G.A (Solve real world and mathematical problems involving area, surface area, and volume) to work in 8th grade. Unit Summary states, “Throughout the geometry standards in sixth grade through eighth grade, students will encounter increasingly complex and multi-part geometric measurement problems, culminating in eighth grade” when students solve problems involving volume of cylinders, cones, and spheres. (8.G.9)

Materials relate grade-level concepts from Grade 6 explicitly to prior knowledge from earlier grades. These references can be found within materials in the Unit Summary or within Lesson Tips for Teachers. Examples include:

• In Unit 2, Unit Rates and Percents, Lesson 6, Tips for Teachers connects 6.RP.3d (Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities) to the work from fourth and fifth Grade. It states, “In Lesson 6, students re-engage in 4.MD.1, 4.MD.2, and 5.MD.1 to make sense of units and their relative sizes” as they solve unit conversion problems while reasoning with measurement units.

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 10, Tips for Teachers connects 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm) to the work from fifth grade (5.NBT.6). It states, “In fifth grade, students found whole number quotients with up to 2 digit divisors and 4 digit dividends using various strategies. In sixth grade, dividing multi-digit whole numbers is a fluency expectation. This lesson reviews partial quotient strategies from fifth grade and students are guided toward understanding the standard algorithm as an efficient way to compute.” In Lesson 10, students divide multi-digit whole numbers using the standard algorithm.

• In Unit 4, Rational Numbers, Unit Summary connects 6.NS.C (Apply and extend previous understandings of numbers to the system of rational numbers) to the work from fifth grade (5.G.A). It states, “In fifth grade, students look at the first quadrant of the coordinate plane and represent locations using ordered pairs of positive numbers. In sixth grade, students build on and extend these concepts to include negative values.” In this unit, “Students are introduced to integers and rational numbers, extending the number line to include negative values, understanding the order of rational numbers, and interpreting them in context.”

• In Unit 7, Geometry, Unit Summary connects 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume) to the work from fifth grade (5.MD.C). It states, “In fifth grade, students explored volume as a measurement of a three-dimensional solid with whole-number side lengths. In this unit, students will reinvestigate how to find volume when packing solids now with fractional unit cubes.”