4th Grade - Gateway 1

The materials reviewed for Fishtank Plus Math Grade 4 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 2, Multi-Digit Multiplication, Post-Unit Assessment, Problem 3 states, “Steph has 40 rubber bands. She has 5 times as many rubber bands as Jake has. Which equation shows how to find the number of rubber bands Jake has?  A. 40 + 5 = 45; B. 40 - 5 = 35; C. 40 × 5 = 200; D. 40 ÷ 5 = 8.” (4.OA.1)

• In Unit 3, Multi-Digit Division, Post-Unit Assessment, Problem 4 states, “In one year, Janie sent 4,368 text messages. Janie sent 4 times as many text messages as Tanner. How many text messages did Tanner send?” (4.NBT.6, 4.OA.2)

• In Unit 5, Fraction Operations, Post-Unit Assessment, Problem 3 states, “Find each sum or difference. a. 6\frac{4}{6}+7\frac{3}{6} b. 5\frac{2}{5}-1\frac{3}{5}.” (4.NF.3c)

• In Unit 6, Decimal Fractions, Post-Unit Assessment, Problem 1 states, “Which of these is equivalent to \frac{38}{100}? A. 0.038; B. 0.38; C. 38.0; D. 380.” (4.NF.6)

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 14, 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 5, 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, Fraction Equivalence and Ordering, Topic C: Comparing and Ordering Fractions,  Lessons 11-15, 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 numerators, 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}.”

• In Unit 8, 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°”

The materials reviewed for Fishtank Plus Math Grade 4 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 5.6 out of 8,  approximately 70%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 101 out of 142, approximately 71%. The total number of lessons includes 134 lessons plus 8 assessments for a total of 142 lessons. 

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work), is 111 out of 154, approximately 72%. There are a total of 20 flex days and 16 of those days are included within units focused on major work, including assessments. By adding 16 flex days focused on major work to the 95 lessons devoted to major work, there is a total of 111 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 101 out of 142, approximately 71%. 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 71% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 4 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 Division, Lesson 11, Problem Set, Problem 7 connects the supporting work of 4.MD.3 (apply the area and perimeter formulas for rectangles in real world and mathematical problems) to the major work 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). It states, “You want to carpet 3 rooms of a house. Using the dimensions below, determine how much carpet is needed: Room 1: Perimeter is 38 yards and the width of the room in 12 yards. Room 2: Perimeter is 50 yards and the width is 13 yards. Room 3: Perimeter is 46 yards and the width is 10 yards. For each room, determine how much carpet is needed.”

• In Unit 4, Fraction Equivalence and Ordering, Lesson 3,  Anchor Task, Problem 1 connects the supporting work of 4.OA.4 (find all the factor pairs for a whole number in the range 1-100) to the major work 4.OA.3 (solve multistep word problems posed with whole numbers and having whole-number answers using the four operations). It states, “Mr. Duffy wants to set up the desks in his room in rows and columns. There are 28 desks in his classroom. What are the different ways he could make rows and columns with 28 desks? Draw arrays to represent the possible arrangements.”

• In Unit 5, Fraction Operations, Lesson 22, Homework, Problem 3 connects the supporting work of 4.MD.4 (make a line plot to display a data set of measurements in fractions of a unit. Solve problems involving addition and subtraction of fractions by using information presented in line plots) to the major work of 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, “Greta recorded the number of miles she walked each day last week on a line plot, as shown below. How many miles in all did Greta walk last week?  A. 8 miles, B. 10 miles. C. 12$$\frac{1}{2}$$ miles, D. 14$$\frac{1}{2}$$ miles.” The line plot includes 1, 1\frac{1}{2}, 2, 2\frac{1}{2}, and 3.

• In Unit 7, Unit Conversions, Lesson 9, Target Task, connects the supporting work of 4.MD.2 (use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit) to the major work of 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). It states, “Solve. Show or explain your work. Marco rode his bike 1 mile. Marco rode four times as far as Allison. Jason rode his bike 5 fewer yards than Allison. How far did Jason ride?”

The materials reviewed for Fishtank Plus Math Grade 4 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, Rounding, Addition, and Subtraction, Lesson 1, Problem Set, Problem 2 connects the major work of 4.NBT.B to the major work of 4.NBT.A as students use place value understanding to round multi-digit whole numbers to any place. Problem 2 states, “Find 412 in your 1,000 book. a. Which hundreds chart is it on? b. Which two tens is it between? c. What would you have to change to make its neighbor to its left?”

• In Unit 2, Multi-Digit Multiplication, Lesson 17, Homework, Problem 4 connects the major work of 4.NBT.B to the major work of 4.OA.A as students use their knowledge of multiplication to solve multi-step word problems. It states, “4. Solve. Show or explain your work. Patti’s sandals weigh 1,167 grams. She bought 3 pairs, all different colors. All 3 pairs of sandals together weigh 239 grams more than her winter boots. What is the weight of Patti’s winter boots?”

• In Unit 3, Multi-Digit Division, Lesson 12, Homework, Problem 2 connects the major work of 4.NBT.B to the major work of 4.OA.A as students find whole-number quotients and remainders in multistep word problems. It states, “Mrs. Turner’s fourth grade class is making gift bags for their 7 parent volunteers. They collected bite sized candy bars which they are distributing equally among each bag. Here is a list of the candy: 37 Dark Chocolate bars, 22 Milky Way bars, 29 Three Musketeer Bars. If Mrs. Turner will keep the leftover candy bars for herself, how many candy bars will Mrs. Turner get?” 

• In Unit 7, Unit Conversions, Lesson 3, Problem Set, Problem 2 connects the supporting work of 4.MD.A to the supporting work of 4.OA.C as students analyze patterns to make the following statements true. It states, “Fill in the blank to make the statement true. a. 4 hundreds 29 ones is ____ones; b. 4 m 29 cm is ____cm; c. 2 thousands 456 ones is ____ones; d. 2 km 456 m is ____m; e. 13 thousands 709 ones is ____ones; f. 13 kg 709 g is ____g; g. ____is 456 thousands 829 ones; h. ____mL is 456 L 829 mL.”

• In Unit 8, Shapes and Angles, Lesson 14, Problem Set, Problem 2 connects the supporting work of 4.MD.C to the supporting work of 4.G.A as students use their ability to measure angles to help classify triangles. It states, “2. △DEF is scalene. What do you observe about its angles? Explain.”

The materials reviewed for Fishtank Plus Math Grade 4 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 2, Multi-Digit Multiplication, Unit Summary states, “Students’ work in this unit will prepare them for fluency with the multiplication algorithm in Grade 5 (5.NBT.5). Students also learn about new applications of multiplication in future grades, including scaling quantities up and down in Grade 5 (5.NF.5), all the way up to rates and slopes in the middle grades (6.RP, 7.RP). Every subsequent grade level depends on the understanding of multiplication and its algorithm, making this unit an important one for students in Grade 4.”

• In Unit 5, Fraction Operations, Unit Summary states, “Students’ understanding of fractions is developed further in Unit 7, in which students explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms (4.NF.5—7). Then, in Grade 5, students extend their understanding and ability with operations with fractions (5.NF.1—7), working on all cases of fraction addition, subtraction, and multiplication and the simple cases of division of a unit fraction by a whole number or vice versa. Students then develop a comprehensive understanding of and ability to compute fraction division problems in all cases in Grade 6 (6.NS.1). Beyond these next few units and years, it is easy to find the application of this learning in nearly any mathematical subject in middle school and high school, from ratios and proportions in the middle grades to functional understanding in algebra.”

• In Unit 6, Decimal Fractions, Unit Summary states, “In Grade 5, students will build on this solid foundation of decimal fractions to develop an even deeper understanding of decimals' relationship to place value and to perform decimal operations with similar models (5.NBT.1—4, 5.NBT.7). By the end of 6th grade, students will be fluent with the use of the standard algorithm to compute with decimals (6.NS.3). From that point forward, students will use their understanding of decimals as a specific kind of number in their mathematical work, including ratios, functions, and many others.”

• In Unit 7, Unit Conversions, Unit Summary states, “But, the unit not only looks back on the year in review, it also prepares students for work in future grades, including the very direct link to converting from smaller units to larger ones in Grade 5 (5.MD.1) and also to ratios and proportions in the middle grades (6.RP.1) as well as many other areas to come.”

Materials relate grade-level concepts from Grade 4 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, Rounding, Addition, and Subtraction, Unit Summary states, “Students’ understanding of the base ten system begins as early as Kindergarten, when students learn to decompose teen numbers as ten ones and some ones (K.NBT.1). This understanding continues to develop in Grade 1, when students learn that ten is a unit and therefore decompose teen numbers into one ten (as opposed to ten ones) and some ones and learn that the decade numbers can be referred to as some tens (1.NBT.1). Students also start to compare two-digit numbers (1.NBT.2) and add and subtract within 100 based on place value (1.NBT.3—5). In second grade, students generalize the place value system even further, understanding one hundred as a unit (2.NBT.1) and comparing, adding, and subtracting numbers within 1,000 (2.NBT.2—9). In Grade 3, place value (NBT) standards are additional cluster content, but they still spend time fluently adding and subtracting within 1,000 and rounding three-digit numbers to the nearest 10 and 100 (3.NBT.1—2).”

• In Unit 3, Multi-Digit Division, Unit Summary states, “Students developed a foundational understanding of division in Grade 3, when they came to understand division in relation to equal groups, arrays, and area. They developed a variety of strategies to build towards fluency with division within 100, and they applied that knowledge to the context of one- and two-step problems using the four operations. Students also came to understand the distributive property, which underpins the standard algorithm for division.”

• In Unit 4, Fraction Equivalence and Ordering, Lesson 1, Tips for Teachers states, “Depending on students’ comfort level with key representations for fractions from Grade 3, particularly tape diagrams/fraction strips and number lines, Lessons 1 and 2 can be combined.” 

• In Unit 6, Decimal Fractions, Unit Summary states, “Students have previously encountered an example of needing to change their understanding of what a number is in Grade 3, when the term came to include fractions. Their Grade 3 understanding of fractions (3.NF.A), as well as their work with fractions so far this year (4.NF.A, 4.NF.B), will provide the foundation upon which decimal numbers, their equivalence to fractions, their comparison, and their addition will be built. Students also developed an understanding of money in Grade 2, working with quantities either less than one dollar or whole dollar amounts (2.MD.8). But with the knowledge acquired in this unit, students will be able to work with money represented as decimals, as it so often is.”

• In Unit 8, Shapes and Angles, Lesson 2, Foundational Skills lists the standard 3.G.A.1 (Understand that shapes in different categories [e.g., rhombuses, rectangles, and others] may share attributes [e.g., having four sides], and that the shared attributes can define a larger category [e.g., quadrilaterals]. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories) as a foundational standard from Grade 3. It is also noted that this skill is covered in the unit.