3rd Grade - Gateway 1

The materials reviewed for Fishtank Plus Math Grade 3 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, Part 1, Post-Unit Assessment, Problem 3 states, “Select the two equations that are correct. A. 2 × 9 = 18; B. 12 ÷ 2 = 10; C. 20 ÷ 5 = 4; D. 3 × 3 = 6.” (3.OA.7)

• In Unit 4, Area, Post-Unit Assessment, Problem 2 states, “A patio is in the shape of a rectangle with a width of 8 feet and a length of 9 feet. What is the area, in square feet, of the patio?” (3.MD.7b)

• In Unit 5, Shapes and Their Perimeter, Post-Unit Assessment, Problem 5 states, “Sonia wants to put a fence around her rectangular backyard. Her backyard is 5 meters long and 6 meters wide.  What is the total length of fence, in meters, Sonia needs to place around the play area?” (3.MD.8)

• In Unit 6, Fractions, Post-Unit Assessment, Problem 6 states, “Angela and Jacob are learning about fractions. Jacob says that the fractions \frac{1}{4} and \frac{2}{3} are equivalent. Angela disagrees with him, so Jacob draws a picture to prove his point: Jacob is incorrect. Explain what is wrong with Jacob’s reasoning.” Two images are provided. One small rectangle is divided into two-thirds and a larger rectangle is divided into one-fourth. The two-thirds rectangle is placed above the one-fourth rectangle and appears to have the same amount shaded in. (3.NF.3d)

The materials reviewed for Fishtank Plus Math Grade 3 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 3 by providing Anchor Tasks, Problem Sets, Homework, and Target Tasks for each lesson. Examples include:

• In Unit 2, Multiplication and Division, Part 1, Lesson 7, Anchor Tasks, Problem 1 engages students with extensive work in 3.OA.5 (apply properties of operations as strategies to multiply and divide). It states, “1. Mr. Barron is working with a small group of students. He sets up their seats facing the whiteboard into two rows with three chairs in each row. a. Draw the seats in Mr. Barron’s room below, using squares to represent the seats. b. Write a multiplication equation that can be used to find the total number of seats Mr. Barron has in the room. c. Determine the total number of seats in Mr. Barron’s room. 2. Sometimes Mr. Barron wants to use chart paper on the side of his room, which you can see in the diagram above. Imagine students facing Mr. Barron at the chart paper. a. Write a multiplication equation that can be used to find the total number of seats Mr. Barron has in the room. b. Determine the total number of seats in Mr. Barron’s room. 3. What do you notice about #1 and #2? What do you wonder?” 

• In Unit 6, Fractions, Lesson 5, Problem Set, Problem 4 engages students with extensive work in 3.NF.1 (understand a fraction \frac{1}{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction \frac{a}{b} as the quantity formed by a parts of size \frac{1}{b}). It states, “An artist plans a wall in a room. The wall is divided into 6 equal parts so that each part can be painted a different color. The artist starts painting the wall. The parts of the wall that look white are not painted yet.” Students read six statements and answer the following question, “Which statements about the wall are correct? Select the two correct statements. A. Each painted part is \frac{1}{4} of the whole; B. Each painted part is \frac{1}{6} of the whole wall; C. Each painted part is \frac{4}{4} of the whole wall; D. The fraction of the wall that is not yet painted is \frac{1}{6}; E. The fraction of the wall not yet painted is \frac{2}{4} F. The fraction of the wall not yet painted is \frac{2}{6}.” 

• In Unit 7, Measurement, Lesson 6, Anchor Task, Problem 2 engages students with extensive work for 3.MD.1 (tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram). It states, “a. Joey gets home at 3:25 p.m. It takes him 7 minutes to unpack and 18 minutes to have a snack before starting homework. What is the earliest time Joey can start his homework? b. Shane’s family wants to start eating dinner at 5:45 p.m. It takes Shane 15 minutes to set the table and 7 minutes to help put the food out. What time should Shane start his chores so that he’ll be ready to eat at 5:45 p.m.? c. Davis has 3 problems for math homework. He starts at 4:08 p.m. The first problem takes him 5 minutes, and the second takes him 6 minutes. If Davis finishes at 4:23 p.m., how long does it take him to solve the last problem?”

The instructional materials provide opportunities for all students to engage with the full intent of Grade 3 standards through a consistent lesson structure, including Anchor Tasks, Problems Sets, Homework Problems, and Target Tasks. Anchor Tasks 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, Multiplication and Division, Part 2, Lessons 21-23 provide the opportunity for students to engage in with the full intent of 3.OA.9 (identify arithmetic patterns, including patterns in the addition table or multiplication table, and explain them using properties of operations). In Lesson 21, Target Task, Problem 1 states, “Look at the 4 and 8 rows in the multiplication table. What relationship do you see? Why does this relationship exist?” Problem 2 states, “Explain how the multiplication table shows the following relationship, 7 × 8 = (5 × 8) + (2 × 8).” In Lesson 22, Anchor Task, Problem 1 states, “Do you think there are more odd or even products on the multiplication chart? Why?” In Lesson 23, Anchor Tasks, Problem 2 states, “Find the next two numbers in the pattern below. 3, 10, 17, 24, 31, _, _. Why do the values in the pattern alternate between even and odd?”

• In Unit 4, Area, Lessons1-4 provide the opportunity for students to engage with the full intent of 3.MD.5 (recognize area as an attribute of plane figures and understand concepts of area measurement) and 3.MD.6 (measure areas by counting unit squares). In Lesson 1, Problem Set, Problem 1 states, “Use triangle pattern blocks to cover each shape below. Draw lines to show where the triangles meet. Then, write how many triangle pattern blocks it takes to cover each shape.” In Lesson 2, Anchor Tasks, Problem 3 states, “Area can also be measured in square inches, also abbreviated “sq in.” a. Measure the length of the sides of one of the tiles given to you to verify that it is a square inch. b. Find the area, in square inches, of the following figures: c. Build a rectangle with an area of 15 square inches.” In Lesson 3, Homework, Problem 2 states, “Each ◻ is 1 square unit. What is the area of each of the following rectangles?” Four different rectangles are provided within a grid. 

• In Unit 6, Fractions, Lessons 1-9, provide the opportunity for students to engage with the full intent of 3.NF.1 (understand a fraction \frac{1}{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction \frac{a}{b} as the quantity formed by a parts of size \frac{1}{b}). In Lesson 1, Target Task, Problem 2 states, “Build a shape with pattern blocks whose fractional unit is fourths. Then trace the shape below.” In Lesson 3, Problem Set, Problem 3 states, “Leroy made a game board, shown below. Each small square of the game board has the same area. What fraction of the game board is shaded? A. \frac{1}{9}; B. \frac{1}{8}; C. \frac{1}{6}; D. \frac{1}{3}”  Lesson 9, Homework, “Draw a model that could represent \frac{3}{4} or \frac{3}{2}. Explain what the whole is for each fraction.”

The materials reviewed for Fishtank Plus Math Grade 3 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 out of 7, approximately 71%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 98 out of 133, approximately 74%. The total number of lessons includes 126 lessons plus 7 assessments for a total of 133 lessons. 

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work), is 108 out of 145, approximately 74%. 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 93 lessons devoted to major work, there is a total of 108 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 98 out of 133, approximately 74%. 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 74% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 3 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 1, Place Value, Rounding, Addition and Subtraction, Lesson 10, Homework, Problem 1 connects the supporting work of 3.NBT.2 (fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction) to the major work of 3.OA.8 (solve two-step word problems using the four operations). It states, “Use the information in the tables to answer the questions below. a. Estimate the total number of hats sold by both stores. b. What is the actual number of hats sold by both stores? c. Estimate the total number of t-shirts sold by both stores. d. What is the actual number of t-shirts sold by both stores? e. Explain how estimating helps you check the reasonableness of your answers.” Relevant information in the table includes South Shore Target sold 36 T-Shirts and 32 Hats; Fenway Target sold 25 T-Shirts and 7 Hats.  

• In Unit 5, Shapes & Their Perimeter, Lesson 14, Homework, Problem 2 connects the supporting work of 3.G.1 (understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category) to the major work of 3.MD.5 (recognize area as an attribute of plane figures and understand concepts of area measurement). It states, “Color tetrominoes on the grid below: a. To create a square with an area of 64 units. b. Create at least two different rectangles each with an area of 24 square units. You may use the same tetromino more than once.”

• In Unit 5, Shapes & Their Perimeter, Lesson 3, Target Task, Problems 1 and 2 connect the supporting work of 3.MD.8 (solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters) to the major work of 3.OA.8 (solve two-step word problems using the four operations). Problem 1 states, “Alan’s rectangular swimming pool is 10 meters long and 16 meters wide. What is the perimeter?” Problem 2 states, “Lila has a pool with a different shape. a. What is the perimeter of Lila’s pool? b. Lila says her pool has a larger perimeter than Alan’s pool. Is she correct? Explain how you know.”

• In Unit 6, Fractions, Lesson 1, Anchor Task, Problem 1 connects the supporting work of 3.G.2 (partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole) to the major work of 3.NF.1 (understand a fraction as \frac{1}{b}, as the quantity formed by 1 part when a whole is partitioned intob equal parts; understand a fraction \frac{a}{b} as the quantity formed by a parts of size \frac{1}{b}). It states, “George made his younger brother, Neil, hexagonal cookies to celebrate his 6th birthday. One of the cookies is shown below. a.Two of Neil’s friends want to share a cookie. They decide to split the cookie in the following way: Is this a fair share of the cookie? Why or why not? b. Neil and his friends Shantelle and Oscar want to share a cookie. They decide to split the cookie in the following way: Is this a fair share of the cookie? Why or why not? c. Four of Neil’s friends want to share a cookie. They decide to split the cookie in the following way: Is this a fair share of the cookie? Why or why not?” [Note: The hexagons are divided into some fair and some not fair partitions.]

The materials reviewed for Fishtank Plus Math Grade 3 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 3, Multiplication and Division, Part 2, Lesson 3, Target Task, Problem 2 connects the major work 3.OA.B to the major work of 3.OA.D as students apply properties of operations to solve two step word problems involving the four operations. It states, “Marcos solves 24 ÷ 6 + 2 = ____. He says it equals 6. Iris says it equals 3. Show how the position of parentheses in the equation can make both answers true.” 

• In Unit 4, Area, Lesson 7, Anchor Task, Problem 3 connects the major work 3.OA.A to the major work of 3.MD.C by using multiplication to solve the area. It states, “a. The area of the rectangle below is 42 square feet. Find the missing side length. b. Explain how you can use either multiplication or division to solve Part (a).” 

• In Unit 5, Shapes and Their Perimeter, Lesson 2, Problem Set, Problem 3 connects the supporting work of 3.MD.D to the supporting work of 3.G.A as students reason with shapes and their attributes as they recognize and solve for perimeter. It states, “Hugh and Daisy draw the shapes shown below. Measure and label the side lengths in centimeters. Whose shape has a greater perimeter? How do you know?”

• In Unit 5, Shapes and Their Perimeter, Lesson 5, Target Task connects the supporting work of 3.NBT.A to the supporting work of 3.MD.D as students fluently add numbers to find the perimeter of shapes. It states, “Marlene ropes off a square section of her yard where she plants grass. One side length of the square measures 9 yards. What is the total length of rope Marlene uses?”

• In Unit 7, Measurement, Lesson 9, Homework, Problem 7 connects the major work of 3.MD.A to the major work of 3.OA.D as students measure the mass of objects and solve two step word problems involving the four operations. It states, “Jennifer’s grandmother buys carrots at the farm stand, as shown below. She and her 3 grandchildren equally share the carrots. Jennifer uses 2 kilograms of her share of carrots to bake a carrot cake. How many kilograms of carrots does Jennifer have left?” The scale indicates the carrots weigh 28 kg.

The materials reviewed for Fishtank Plus Math Grade 3 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, Place Value, Rounding, Addition and Subtraction, Unit Summary states, “In Grade 4, students learn about multiplicative comparison; i.e.: a value being x times as many as another value. Thus, students’ understanding of the place value system is more precisely refined as “a digit in one place represents ten times what it represents in the place to its right” (4.NBT.1, emphasis added). “Further, students learn to round any multi-digit number to any place. They also use the standard algorithm to solve addition and subtraction problems to the new place values they encounter at this grade level, namely, to one million. Thus, while the majority of the content learned in this unit comes from an additional cluster, they are deeply important skills necessary to fully master the major work of the grade with 3.OA.8, as well as a foundation for rounding and the standard algorithms used to any place value learned in Grade 4 (4.NBT.1—4) and depended on for many grade levels after that.”

• In Unit 4, Area, Unit Summary states, “In future grades, students will rely on the understanding of area to solve increasingly complex problems involving area, perimeter, surface area, and volume (4.MD.3, 5.MD.3—5, 6.G.1—4). Students will also use this understanding outside of their study of geometry, as multi-digit multiplication problems in Grade 4 (4.NBT.5), fraction multiplication in Grade 5 (5.NF.4), and even polynomial multiplication problems in Algebra (A.APR.1) rely on an area model.”

• In Unit 5, Shapes and Their Perimeter, Unit Summary states, “Students will further deepen their understanding of these ideas in future grade levels. In Grade 4, students solve more complex word problems involving area and perimeter (4.MD.3), as well as classify shapes based on the presence of parallel and perpendicular shapes (4.G.2), which is very connected to their study of angles (4.MD.5—7). The beginning work on categorization in Grade 3 culminates in Grade 5, where students have a complete picture of the hierarchical nature of classifying shapes (5.G.3). In the middle grades and high school, increasingly complex problems rely on students’ deep understanding of attributes of shapes and how to measure them, threaded throughout this unit.”

• In Unit 7 Measurement, Unit Summary states, “Students will rely on the work of this unit to convert from a larger unit to a smaller unit in Grade 4 (4.MD.1) and from a smaller unit to a larger one in Grade 5 (5.MD.1), as well as to solve multi-step word problems involving intervals of time, liquid volumes, and masses of objects, including problems involving simple fractions or decimals (4.MD.2, 5.MD.1). Beyond the direct connections to Grade 5 Common Core State Standards, “measurement is central to mathematics, to other areas of mathematics (e.g., laying a sensory and conceptual foundation for arithmetic with fractions), to other subject matter domains, especially science, and to activities in everyday life. For these reasons, measurement is a core component of the mathematics curriculum” (GM Progression, p. 1).”

Materials relate grade-level concepts from Grade 3 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 2, Multiplication and Division, Part 1, Unit Summary states, “In Grade 2, students learned to count objects in arrays using repeated addition (2.OA.4) to gain a foundation to multiplication. They’ve also done extensive work on one- and two-step word problems involving addition and subtraction, having mastered all of the problem types that involve those operations (2.OA.1). Thus, students have developed a strong problem-solving disposition and have the foundational content necessary to launch right into multiplication and division in this unit. At the start of this unit, students gain an understanding of multiplication and division in the context of equal group and array problems in Topic A. To keep the focus on the conceptual understanding of multiplication and division (3.OA.1, 3.OA.2), Topic A does not discuss specific strategies to solve, and thus students may count all objects (a Level 1 strategy) or remember their skip-counting and repeated addition (Level 2 strategies) from Grade 2 to find the product. In Topics B and C, however, the focus turns to developing more efficient strategies for solving multiplication and division, including skip-counting and repeated addition (Level 2 strategies) as well as ‘just knowing’ the facts, which works toward the goal that ‘by the end of grade 3, [students] know from memory all products of two single-digit numbers and related division facts’ (3.OA.7).” 

• In Unit 4, Area, Lesson 5, Tips for Teachers states, “Today’s lesson is the first one where students will not be given concrete units in order to find the area of rectangles. Students may still want to draw individual units to find the area of rectangles, but hopefully most students are completing rows and columns instead. As the Progressions note, “less sophisticated activities of this sort were suggested for earlier grades so that Grade 3 students begin with some experience”, so development towards this row and column understanding should be fairly straightforward (GM Progression, p. 17).”

• In Unit 7, Measurement, Lesson 8, Foundational Standards lists 2.MD.A.1 (Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes) and 2.MD.B.6 (Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram) as Foundational Standards for this lesson. 

• In Unit 6, Fractions, Unit Summary, “In Unit 6, students extend and deepen Grade 1 work with understanding halves and fourths/quarters (1.G.3) as well as Grade 2 practice with equal shares of halves, thirds, and fourths (2.G.3) to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line.”