5th Grade - Gateway 2

The instructional materials reviewed for Zearn Grade 5 meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific cluster headings, such as 5.NBT.A, 5.NF.B, and 5.MD.C.

Students expand their understanding of fractions to include addition and subtraction of fractions with unlike denominators (5.NF.A) as well as multiplication and division of fractions for limited cases (5.NF.B).

• In Mission 3, Small Group Lesson 3, students develop a conceptual model of fraction addition. Students fold pieces of paper and shade sections of the paper to represent each fraction that will be added together.
• In Mission 4, Small Group Lesson 6, students develop an understanding of finding a fraction of a set. Students use a tape diagram to show an understanding of the solution to the following problem, “In a class of 24 students, ⅚ are boys. How many boys are in the class?”
• In Mission 4, Small Group Lesson 25, students develop a conceptual model of division with fractions. Students are asked to solve and model the following problem: “Tien wants to cut ¼ foot lengths from a board that is 5 feet long, How many board can he cut?”

The use of multiple representations and understanding the similarities and differences between representations is used extensively throughout the instructional materials to help students build conceptual understanding.

• Mission 1 focuses on 5.NBT.A. Place value charts and vertical number lines are used to introduce the fractional unit of thousandths as well as place value understanding to round a decimal to any place.
• Mission 5 focuses on 5.MD.C. Students use centimeter grid paper, centimeter cubes, and isometric dot paper to build and represent solid figures of a given volume as well as to find the volume of a given rectangular prism.
• Throughout the Grade 5 Missions and Lessons, students are frequently asked to draw and make conclusions based on their drawings.
• Optional Problem Sets include opportunities for students to draw pictures and make models to show understanding. For example, the Mission 4 Problem Set for Small Group Lesson 16, asks students to, “Solve and show your thinking with a tape diagram” when multiplying fractions within a word problem.

Overall, Lessons within Missions, whether Teacher-Led Instruction or Independent Digital Lessons, present opportunities for students to develop conceptual understanding of the mathematical concepts for the grade using place value, concrete models, and the properties of arithmetic.

The instructional materials reviewed for Grade 5 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Missions address procedural skill and fluency in both the Independent Digital Lessons, with Fluency activities titled Number Gym, Sprint, Blast, Totally Times, and Multiply Mania, and in Small Group Instruction, with Fluency activities for most lessons.

• In Mission 2, Teacher-Led Instruction Whole Group Fluency Lesson 4 (5.NBT.5), students practice decomposing multiplication sentences such as 12 x 3 =____ into (8 x 3) + (____ x 3) = _____. They practice writing the value of the new equation and using addition to solve. They repeat the process for 14 x 4, 13 x 3, and 15 x 6.
• In Mission 2, Independent Digital Lesson 5 Blast, students practice solving two-digit whole numbers multiplied by 10. During Blast, students solve multi-digit multiplication (5.NBT.5).

Overall, Zearn includes time in every lesson during Independent Digital Lessons in Number Gyms and lesson-specific activities to build fluency. Most Teacher-Led Instruction lessons include a Whole Group Fluency lesson, as well. These lessons are designed to complement one another, reinforcing student development of procedural skills and fluency.

The instructional materials reviewed for Grade 5 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

During Teacher-Led Instruction in every Mission, there are Whole Group Word Problems (Application Problems) for most lessons. These Application Problems represent the Addition and Subtraction Situations described in Table 1 of the CCSSM, and the Multiplication and Division Situations described in Table 2 of the CCSSM.

Mission 4, Multiply and Divide Fractions and Decimals (5.NF.B) represents major work for the grade. The Application Problems in this mission are specifically designed as a bridge between deepening concept development on multiplication and division and applying their understanding to multiplication and division of fractions. For example, in Teacher-Led Instruction, Whole Group Word Problems, Topic E: Multiplication of a Fraction by a Fraction, “Topic E introduces students to multiplication of fractions by fractions -both in fraction and decimal form (5.NF.4a, 5.NBT.7).” The topic includes six lessons that have application problems. For example:

• Lesson 15: “Kendra spent ⅓ of her allowance on a book and ⅖ on a snack. If she had four dollars remaining after purchasing a book and snack, what was the total amount of her allowance?” The teacher note states, “This problem reaches back to addition and subtraction of fractions, as well as fractions of a set.”
• Lesson 17: “Ms. Casey grades 4 tests during her lunch. She grades ⅓ of the remainder after school. If she still has 16 tests to grade after school, how many tests are there?” The teacher note states, “Today’s Application Problem recalls Lesson 16’s work with tape diagrams.”
• Lesson 18: “An adult female gorilla is 1.4 meters tall when standing upright. Her daughter is 3 tenths as tall. How much more will the young female gorilla need to grow before she is as tall as her mother?” The teacher note states, “This Application Problem reinforces that multiplying a decimal number by tenths can be interpreted in fraction or decimal form (as practiced in Lesson 17). Students who solve this problem by converting to smaller units (centimeters or millimeters) should be encouraged to compare their process to solving the problem using 1.4 meters.”

Throughout Grade 5 students deepen their understanding of multiplication and division as they practice multiplication and division of fractions. The Application Problems link the four operations of arithmetic and the properties of arithmetic to major work of the grade.

The instructional materials reviewed for Grade 5 meet the expectation for balancing the three aspects of rigor. Overall, the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within Teacher-Led Instruction and Independent Digital Lessons.

In each Mission students develop procedural skills and fluency and conceptual understandings, and apply these to solve real-world problems.

• Fluency is embedded into every Lesson. Mission 2, Teacher-Led Instruction, Whole Group Fluency Lesson 14 builds on understandings from Grade 4 to help students review the multiplication of unit fractions. In Independent Digital Lessons Mix and Match, students practice multiplication of multi-digit whole numbers.
• Conceptual understanding is consistently embedded into lessons. In Mission 3, Teacher-Led Instruction, Lesson 8, students use the number line when adding and subtracting fractions greater than or equal to 1. During Independent Digital Lessons, Tower of Power Lesson 8, students use the number line when adding and subtracting mixed numbers and whole numbers.
• Application problems are consistently embedded into lessons and often call for students to model their thinking and make connections to procedural skills. The Mission 5, Teacher-Led Instruction, Whole Group Word Problems Lesson 3 states: “An ice cube tray has two rows of 8 ice cubes. How many ice cubes are in a stack of 12 ice cube trays? Draw a picture to explain your reasoning.”.

The instructional materials reviewed for Zearn Grade 5 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. The students’ materials in the Small Group Lessons, Whole Group Word Problems, Optional Problem Sets, and Assessments provide opportunities throughout the year for students to both construct viable arguments and analyze the arguments of others. The students’ materials sometimes prompt students to construct viable arguments and include some opportunities for students to analyze the arguments of others.

Students are asked daily to explain their thinking while completing application problems. MP.3 is identified through Whole Group Word Problems, Whole Group Fluency, and Assessment. Examples of opportunities to analyze the arguments of others:

• Mission 3, Teacher-Led Instruction, Optional Problem Set, Lesson 5, Question 3: “Sandra says that 4/7 - ⅓ = ¾ because all you have to do is subtract the numerators and subtract the denominators. Convince Sandra that she is wrong. You may draw a rectangular fraction model to support your thinking.”
• Mission 5, Teacher-Led Instruction, Optional Problem Set, Lesson 1, Problem 3: “Joyce says that the figure below, made of 1 cm cubes, has a volume of 5 cubic centimeters. Explain her mistake.” Students explain that Joyce didn’t count a hidden unit on the first layer of the solid.
• Mission 6, Mid-Module Assessment, Question 4: “Garrett and Jeffrey are planning a treasure hunt. They decide to place a treasure at a point that is a distance of 5 units from the x-axis and 3 units from the y-axis. Jeffrey places a treasure at point J and Garrett places one at point G. Who put the treasure in the right place? Explain how you know.”

Examples of opportunities to construct viable arguments:

• Mission 2, End-of-Module Assessment, Question 2, Parts a and b: Students explain their thinking as they determine the estimated quotient by rounding the expression to relate to a one-digit fact.
• Mission 3, Teacher-Led Instruction, Optional Homework, Lesson 3, Questions 2 - 4: Students are instructed to “Solve the following problems. Draw a picture, and write the number sentence that proves the answer. Simplify your answer, if possible.”
• Mission 4, End-of-Module Assessment, Question 5, Part d: Students create a viable argument when asked, “Could either of the problems also be solved by using ½ x 5? If so, which one(s)? Explain your thinking.”
• Mission 6, Teacher-Led Instruction, Optional Problem Set, Lesson 5, Question 7: Students construct an argument from a given list of coordinate pairs for what Adam should guess next in order to sink Janice’s battleship. The problem reads, “Adam and Janice are playing Battleship. Presented in the table is a record of Adam’s guesses so far. He has hit Janice’s battleship using these coordinate pairs. What should he guess next? How do you know? Explain, using words and pictures.”

The Zearn Grade 5 materials meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Overall, there is guidance for teachers on how to lead student discussions in which students construct their own viable arguments and analyze the arguments of others.

The Teacher-Led Instruction Lessons provide opportunities for teachers to discuss the mathematics with their students and for students to discuss the mathematics with each other, as directed by the teacher. For example:

• In Mission 1, Teacher-Led Instruction, Small Group, Lesson 13, Problems 4-6, teachers are instructed to, “Have students decompose the decimal several ways and then reason about which is the most useful for division. It is also important to draw parallels among the next three problems. Lead students by asking questions such as, 'How does the answer to the second set of problems help you find the answer to the third?' if necessary.”
• In Mission 2, Teacher-Led Instruction, Small Group, Lesson 28, Problem 3, teachers give students a problem that involves cutting a board into multiple pieces of different lengths. After students solve the problem, the teacher asks, “How can you be sure your final answer is reasonable?” and “How did you organize your work so that you could keep track of all the different steps? Compare your organization with that of your partner.” These prompts require students to construct an argument for their process as well as analyze their partner’s process to finding a solution.
• In Mission 5, Teacher-Led Instruction, Small Group, Lesson 16, Problem 2, students compare the trapezoid that they created when they added a third and fourth segment that crossed a set of parallel lines. The teacher instructs the students to, “Compare your trapezoid with your partner’s. What is alike? What is different?”
• In Mission 6, Teacher-Led Instruction, Small Group, Lesson 7, teachers provide guidance for students to construct an argument by stating, “Tell a neighbor how you know.” Students critique the reasoning of their peers when the teacher states, “Some of you said the rule for the coordinate pair is x times 2, and some of you said that rule is y is always 3. Which relationship is correct? How do you know? Turn and talk.”

The instructional materials reviewed for Zearn Grade 5 meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics that is present throughout.

The instructional materials provide instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers in Teacher-Led Instruction and Independent Digital Lessons.

• Vocabulary is used directly in the Teacher-Led Instruction Small Group Lessons and then reinforced in the Whole Group Word Problems. Teachers, when applicable, model the vocabulary. For example, Mission 6, Teacher-Led Instruction, Whole Group Word Problems, Lesson 2 states, “The Application Problem prepares students for today’s discussions regarding parallel and perpendicular lines.”
• Vocabulary is sometimes explicitly taught during the Guided Practice part of the Independent Digital Lessons. Vocabulary words are in bold and explained and are followed up by models or examples. For example, Mission 5, Independent Digital Lesson 1, Math Chat introduces students to the terms, cubic centimeter and volume, and has students complete several examples identifying the volume of solids in cubic centimeters.
• Students are expected to use correct mathematics vocabulary as they Read, Draw, and Write for Word Problems. For example, in Mission 2, Teacher-Led Instruction, Whole Group Word Problems, Lesson 14, students must use correct terminology and representations as they draw and label a tape diagram to represent a given unit within a measurement system. Problem 3 reads, “Express 1 quart as a fraction of 1 gallon.”