5th Grade - Gateway 1

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into seven 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 2, Multiplication and Division of Whole Numbers, Post-Unit Assessment, Problem 3 states, “Which expression matches the statement, ‘the sum of 2 and 4 subtracted from 9’? A. 2 + 9 – 4; B. 9 – 2 + 4; C. 9 – (2 + 4); D. (2 + 4) – 9.” (5.OA.2)

• In Unit 5, Multiplication and Division of Fractions, Post-Unit Assessment, Problem 4 states, “Jin had 60 stickers in her collection. She gave $$\frac{3}{5}$$ of the stickers to her friend. How many stickers did Jin give to her friend? A. 12; B. 20; C. 36; D. 40.” (5.NF.4)

• In Unit 6, Multiplication and Division of Decimals, Post-Unit Assessment, Problem 6 states, “a. If 358 × 25 = 8,950, then what is 3.58 × 25 equal to? Explain your reasoning. b. If 12 ÷ 2 = 6, then what is 120 ÷ 0.2 equal to? Explain your reasoning.” (5.NBT.7)

• In Unit 7, Patterns and Coordinate Plane, Post-Unit Assessment, Problem 1 states, “Which of the following correctly describes a way to plot the point (2, 5) on a coordinate plane? A. Start at the origin. Move 2 units up the 𝑦-axis, then move 5 units to the right. Plot the point there.; B. Start at the top of the 𝑦-axis. Move 2 units down the 𝑦-axis, and then move 5 units to the right. Plot the point there.; C. Start at the origin. Move 2 units to the right on the 𝑥-axis, and then move 5 units up. Plot the point there.; D. Start at the top of the 𝑦-axis. Move 2 units to the right, and then move 5 units down. Plot the point there.” (5.G.1)

The materials reviewed for Fishtank Plus Math Grade 4 meet expectations for 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 4 by providing Anchor Tasks, Problem Sets, Homework, and Target Tasks for each lesson. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 19, Target Task engages students in extensive work in 4.OA.3 (solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). It states, “In one year a factory used 11650 meters of cotton, 4,950 fewer meters of silk than cotton, and 3,500 more meters of wool than silk. 1. How many meters in all were used in the three fabrics? Show or explain your work. 2. Assess the reasonableness of your answer.” 

• In Unit 2, Multi-Digit Multiplication, Lesson 19, Problem Set, Problem 2 engages students in extensive work in 4.NBT.5 (multiply a whole number of up to four digits by a one-digit whole number, and multiply two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). It states, “2. Use the same method as Felicia to complete an area model and an equation to solve each of the following multiplication problems. a. 14 × 22 b. 25 × 32.” 

• In Unit 6, Fraction Operations, Lesson 12, Homework, Problems 1-3 engage students in extensive work in 4.NF.3c (add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction). It states, “1. Solve. Show or explain your work. a. 4\frac{1}{3}+\frac{1}{3} b. 4\frac{1}{4}+\frac{2}{4} c. \frac{2}{6}+3\frac{4}{6} d. \frac{5}{8}+7\frac{3}{8} 2. Find the sum in two ways. 5\frac{7}{10}+\frac{4}{10}; 3. Georgia was solving #2, and wrote 5\frac{11}{10} as her answer. How would you suggest Georgia change the way she has recorded her answer? Why is it helpful to record answers that way?” 

The instructional materials provide opportunities for all students to engage with the full intent of Grade 4 standards through a consistent lesson structure, including Anchor Tasks, Problems Sets, Homework Problems, 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. Each unit is further divided into topics. The lessons within each topic build on each other, meeting the full intent of the standards. Examples of where the materials meet the full intent include:

• In Unit 3, Multi-Digit Division, Lesson 9 provides an opportunity for students to engage with the full intent of standard 4.NBT.6 (find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models).  Target Task, Problem 2 is written and solved using the standard algorithm. It states, “Here is a calculation of 8,472 ÷ 5. a. There’s a 5 under the 8 in the 8,472. What does this 5 represent? b. What does the subtraction of 5 from 8 mean? c. Why is a 4 written next to the 3 from 8-5?” Problem set, Problem 2 states, “Write a division problem whose quotient is 314 R 7. Explain how you came up with it.”  Homework, Problem 4 states, “Tamieka is making bracelets. She has 3,467 beads. It takes 8 beads to make each bracelet. How many bracelets can she make? How many more beads would she need to be able to make another bracelet?” 

• In Unit 4, Shapes and Angles, Topic B: Measures of Angles, Lessons 5-10 provide an opportunity for students to engage with the full intent of standards 4.MD.5 (recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement) and 4.MD.6 (measure angles in whole-number degrees using a protractor. Sketch angles of specified measure). In Lesson 7, Problem Set, Problem 1 states, “Do not use your protractor to solve! a. Look at the angle shown. Which measure is closest to the measure of the angle? a. 140 b. 90 c. 40. d. 15.” In Lesson 8, Problem Set, Problem 2 states, “Maria says that this angle measures 153 degrees. Is she correct or incorrect? Why.” Lesson 9, Anchor Tasks, Problem 2 states, “Sketch angles that have each of the following angle measures. a. 80° b. 133°”

• In Unit 5, Fraction Equivalence and Ordering, Topic B: Comparing and Ordering Fractions,  Lessons 7-11, provides an opportunity for students to engage with the full intent of 4.NF.2 (compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numberators, or by comparing to a benchmark fraction such as \frac{1}{2}. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >,=, or <, and justify the conclusions, e.g., by using a visual fraction model).  In Lesson 7, Problem Set, Problem 2 states, “Compare each pair of fractions using >, <, or =. a. \frac{3}{4} ___ \frac{3}{7} b. \frac{2}{5} ___ \frac{4}{9} c. \frac{2}{3} ___ \frac{5}{6} d. \frac{3}{8} ___ \frac{1}{4} e. \frac{7}{11} ___ \frac{7}{13} f. \frac{8}{9} ___ \frac{2}{3} g. \frac{2}{3} ___ \frac{5}{6} h. \frac{3}{4} ___ \frac{7}{12}.” Problem 5 states, “Select True if the comparison is true. Select False if the comparison is not true. \frac{89}{100}>\frac{9}{10}; \frac{7}{12}<\frac{2}{3};  \frac{3}{5}>\frac{4}{10}.” In Lesson 9, Homework, Problem 3 states, “Rowan has 3 pieces of yarn, as described below. A red piece of yarn that is \frac{3}{4} foot long; A yellow piece of yarn that is \frac{6}{8} foot long; A blue piece of yarn that is \frac{4}{12} foot long. Which number sentence correctly compares the lengths of 2 of these pieces of yarn? A. \frac{3}{4}<\frac{6}{8}; B. \frac{4}{12}<\frac{3}{4}; C. \frac{3}{4}>\frac{6}{8}; D. \frac{4}{12}>\frac{6}{8}.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major work 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 5.1 out of 7,  approximately 73%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 99 out of 129, approximately 77%. The total number of lessons includes 122 lessons plus 7 assessments or a total of 129 lessons. 

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work), is 109 out of 141, approximately 77%. There are a total of 19 flex days and 15 of those days are included within units focused on major work, including assessments. By adding 15 flex days focused on major work to the 94 lessons devoted to major work, there is a total of 109 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 99 out of 129, approximately 77%. 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 77% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 5 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 2, Multiplication and Division of Whole Numbers, Lesson 20, Homework, Problem 3 connects the supporting work of 5.OA.1 (use parentheses, brackets, or braces in numerical expression, and evaluate expressions with these symbols) to the major work of 5.NBT.5 (fluently multiply multi-digit whole numbers using the standard algorithm). It states, “Frances is sewing a border around 2 rectangular tablecloths that each measure 9 feet long by 6 feet wide. If it takes her 3 minutes to sew on 1 inch of border, how many minutes will it take her to complete her sewing project? Write an expression, and then solve.”

• In Unit 3, Shapes and Volume, Lesson 5, Anchor Task, Problem 3 connects the supporting work of 5.OA.2 (write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). It states, “Akiko and Philip are finding the volume of the following rectangle prism. Philip says that you have to multiply length by width by height, so you have to multiply 10 × 14 × 2. Akiko says the computation will be easier if you multiply 10 × 2 × 14. a. Is Philip correct? Must the dimensions be multiplied in that order? Show or explain your thinking. b. Why do you think Akiko thinks that multiplying 10 × 2 × 14 will be an easier computation? Is it possible to multiply the dimensions in that order? Show or explain your thinking. c. Use what you’ve concluded from Parts (a) and (b) to explain how you would calculate the volume of a rectangular prism whose length is 4 feet, width is 7 feet, and height is 15 feet.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 16, Problem Set, Problem 1 connects the supporting work of 5.OA.2 (write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to the major work of 5.NF.4 (apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction). It states, “The total distance around a running track is $$1\frac{5}{8}$$ miles. Wayne ran $$\frac{1}{4}$$ of the track. Which of the following equations can be used to find d, the distance in miles that Wayne ran? a. $$\frac{1}{4}×\frac{13}{8}$$, b. $$\frac{1}{4}×\frac{13}{8}$$, c. $$\frac{4}{1}×\frac{13}{8}$$, d. $$\frac{4}{1}×\frac{15}{8}$$”.

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Target Task, Problem 2  connects the supporting work of 5.MD.1 (convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-stop, real world problems) with the major work of 5.NF.4 (apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). It states, “Solve. Show or explain your work. Yolanda took a bus to visit her grandmother. She brought a CD to listen to on the bus. The CD is 78 minutes long. The bus ride was $$2\frac{1}{2}$$ hours long. How many minutes longer was the bus ride than the CD?”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples of connections from supporting work to supporting work and/or from major work to major work throughout the grade-level materials, when appropriate, include:

• In Unit 1, Place Value with Decimals, Lesson 9, Target Task, Problem 1 connects the major work of 5.NBT.A to the major work of 5.NF.B as students write numbers with decimal values in expanded form and use whole numbers multiplied by the fractional value of the place (e.g., 4 x 110).  It states, “A number is given below. 136.25 In a different number, the 6 represents \frac{1}{10} of the value of the 6 in the number above. What value is represented by the 6 in the other number? A. Six hundredths B. Six tenths C. Six ones D. Six tens.”

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 4, Anchor Tasks, Problem 2 connects the major work of 5.NBT.A to the major work on 5.NBT.B as students apply their understanding of the value of a digit in multiplying numbers. It states, “Solve.

  1. 60 × 5 = ___

  2. 60 × 50 = ___

  3. 60 × 500 = ___

  4. 60 × 5,000 = ___.” 

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 13, Anchor Tasks, Problem 2 connects the major work of 5.NBT.B to the major work of 5.NBT.A, as students perform operations with multi-digit whole numbers and decimals to hundredths, and understand the place value system. It states, “Solve. Show your work with an area model. a. 0.3 + 0.5 b. 0.64 + .07.”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Tasks connects the supporting work of 5.G.A to the supporting work 5.OA.B as students represent real world problems within the first quadrant of a coordinate plane, and generate patterns using rules. It states, “Jessica has $15 saved. She earns $6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs $45. How many hours does Jessica need to babysit in order to be able to buy it?” 

• In Unit 7, Patterns and the Coordinate Plane, Lesson 12, Homework, Problem 5 connects the supporting work of 5.MD.B to the supporting work of 5.G.A as students enter data in a table, interpret the data, and complete a line graph on a coordinate plane. It states, “5. Use the following two patterns to complete Parts (a)—(d): Pattern for x-coordinates: Start at 1, add 4; Pattern for y-coordinates: Start at 3, add 4; A. Complete the following table. B. Plot each point on the coordinate plane to the right. C. Use a straightedge to construct a line through these points. D. Give the coordinates of two other points that fall on this line with the x-coordinates greater than 25. (___, ___ ) and (___, ___) E. How did you find two other points that would lie on this line?”

The materials reviewed for Fishtank Plus Math Grade 5 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 3, Shapes and Volume, Unit Summary states, “In Grade 6, students will explore concepts of length, area, and volume with more complex figures, such as finding the area of right triangles or finding the volume of right rectangular prisms with non-whole-number measurements (6.G.1, 6.G.2). Students will even rely on their understanding of shapes and their attributes to prove various geometric theorems in high school (GEO.G-CO.9—11). Thus, this unit provides a nice foundation for connections in many grades to come.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Unit Summary states, “As previously mentioned, students will explore the other operations, multiplication and division, of fractions and decimals in Units 5 and 6, including all cases of fraction and decimal multiplication and division of a unit fraction by a whole number and a whole number by a unit fraction (5.NF.3–7, 5.NBT.7). In Grade 6, students encounter the remaining cases of fraction division (6.NS.1) and solidify fluency with all decimal operations (6.NS.3). Students then rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals.)”

• In Unit 5, Multiplication and Division of Fractions, Unit Summary states, “In Grade 6, students encounter the remaining cases of fraction division (6.NS.1). Work with fractions and multiplication is a building block for work with ratios. In Grades 6 and 7, students use their understanding of wholes and parts to reason about ratios of two quantities, making and analyzing tables of equivalent ratios, and graphing pairs from these tables in the coordinate plane. These tables and graphs represent proportional relationships, which students see as functions in Grade 8” (NF Progression, p. 20). Students will further rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals.

• In Unit 7, Patterns and The Coordinate Plane, Unit Summary states, “This work is an important part of ‘the progression that leads toward middle-school algebra’ (6—7.RP, 6—8.EE, 8.F) (K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics, p. 7). This then deeply informs students’ work in all high school courses. Thus, Grade 5 ends with additional cluster content, but that designation should not diminish its importance this year and for years to come.”

Materials relate grade-level concepts from Grade 5 explicitly to prior knowledge from earlier grades. These references can be found within materials in the Unit Summary, within Lesson Tips for Teachers, and in the Foundational Skills information in each lesson. Examples include:

• In Unit 1, Place Value with Decimals, Unit Summary states, “In Grade 4, students developed the understanding that a digit in any place represents ten times as much as it represents in the place to its right (4.NBT.1). With this deepened understanding of the place value system, students read and wrote multi-digit whole numbers in various forms, compared them, and rounded them (4.NBT.2—3).”

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 5, Foundational Skills lists the standards 4.NBT.B.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm) and 4.NBT.B.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) as foundational standards from Grade 4.  It is also noted that these skills are covered in the unit.

• In Unit 2, Multiplication and Division of Whole Numbers, Unit Summary states, “In Grade 4, students attained fluency with multi-digit addition and subtraction (4.NBT.4), a necessary skill for computing sums and differences in the standard algorithm for multiplication and division, respectively. Students also multiplied a whole number of up to four digits by a one-digit whole number, as well as two two-digit numbers (4.NBT.5). By the end of Grade 4, students can compute those products using the standard algorithm, but ‘reason repeatedly about the connection between math drawings and written numerical work, help[ing] them come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities’ (Progressions for the CCSSM, ‘Number and Operation in Base Ten, K-5’, p. 14). Students also find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (4.NBT.6). Similar to multiplication, by the end of Grade 4, students can compute these quotients using the standard algorithm alongside other strategies and representations so that the algorithms are meaningful rather than rote.”

• In Unit 5, Multiplication and Division of Whole Numbers, Tips for Teachers states, “Students have seen the area model, the partial products algorithm, and the standard algorithm strategies with two-digit by two-digit multiplication in Grade 4.”

• In Unit 6, Multiplication and Division of Decimals, Unit Summary states, “In Grade 4, students were first introduced to decimal notation for fractions and reasoned about their size (4.NF.5—7). Then, in the first unit in Grade 5, students developed a deeper understanding of decimals as an extension of our place value system, understanding that the relationships of adjacent units apply to decimal numbers, as well (5.NBT.1), and using that understanding to compare, round, and represent decimals in various forms (5.NBT.2—4). Next, students learned to multiply and divide with whole numbers in Unit 2 (5.NBT.5—6), skills upon which decimal computations will rely.”