3rd Grade - Gateway 2

The instructional materials reviewed for Zearn Grade 3 meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings, such as 3.OA.A, 3.OA.B and 3.NF.A.

Cluster 3.OA.B states that students are to understand properties of multiplication and the relationship between multiplication and division.

• Mission 1 uses pictures and arrays to develop conceptual understanding of multiplication and division. In Mission 1, Teacher-Led Instruction, Lesson 1 students use their arms to relate skip counting to repeated addition to ten groups of two.
• Missions 1 and 3 use counters, tape diagrams, arrays, and drawings to help visualize division and connect it to multiplication. In Mission 3, Teacher-Led Instruction, Lesson 6 students model the distributive property on whiteboards and tape diagrams to show multiplication of 6 and 7. Using the “break apart” method, students break larger numbers into smaller numbers to compute the product of 6 x 7, arriving at different ways to demonstrate the distributive property (i.e. 6x4 + 6x3 or 6x5 + 6x2).

Cluster 3.NF.A states students will develop understanding of fractions as numbers. In Mission 5, Fractions as Numbers, the lessons use concrete models to demonstrate partitioning shapes into equal parts.

• In Teacher-Led Instruction, Lesson 9 students use drawings and fractions strips to model how an orange can be partitioned into fourths. Students explore questions such as the following:
  • “What is our unit?"
  • "How many fourths are in two oranges?"
  • "Are you sure our unit is still fourths? Talk with your partner.”
• These questions and prompts help students articulate the relationship of parts to whole and what happens when there is more than one whole, as in this case with two oranges with a unit of one-fourth of one orange.

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. Students use place value concepts to round numbers and perform multi-digit addition and subtraction within 1,000 (as described in Table 1 of the CCSSM). Students link multiple representations of multi-digit whole numbers to place value charts and decomposing and recomposing of numbers with hands-on and virtual manipulatives, and these representations are connected to addition and subtraction sentences within 1,000. Students develop an understanding of fractions as numbers by exploring fractional parts of a whole and the relationship between unit fractions and their whole.

The instructional materials reviewed for Grade 3 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.

• Mission 2, Teacher-Led Instruction Whole Group Fluency includes activities (3.NBT.2) where students use part-whole thinking with metric units (Lesson 15), count forward and backwards by nines to 90 (Lesson 17), and use the subtraction algorithm within 1000 with various units (Lesson 19).
• In Mission 2, Independent Digital Lessons students add measurements using the standard algorithm to compose larger units and subtract measurements including three-digit minuends.

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 3 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. For example, Cluster 3.OA.A represents major work of the grade. Mission 1, Whole Group Word Problems for Topic B, "Division as an Unknown Factor Problem" connects multiplication and division situations to different applications from Lessons 4 through 6. For example:

• The Lesson 4: “The student council holds a meeting in Mr. Chang’s classroom. They arrange the chairs in 3 rows of 5. How many chairs are used in all?” The teacher note states, “This problem reviews relating multiplication to the array model from Lesson 2. This application word problem connects multiplication to a division situation asking how many in each group?“
• The Lesson 5: “Stacey has 18 bracelets. After she organizes the bracelets by color, she has 3 equal groups. How many bracelets are in each group?” The teacher note states, “This problem reviews the meaning of the unknown as the size of the group in division from Lesson 4. It also provides a comparison to Problem 1 in the Concept Development, where the unknown represents the number of groups in division.”
• The Lesson 6: “Twenty children play a game. There are 5 children on each team. How many teams play the game? Write a division sentence to represent the problem.” The teacher note states, “... where the unknown represents the number of groups.”

Throughout Grade 3 students develop their understanding of multiplication and division through a variety of multi-digit addition and subtraction. The Application Problems link the four operations of arithmetic to the CCSSM Tables 1 and 2 and to the major work of the grade.

The instructional materials reviewed for Grade 3 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. In Mission 3, Independent Digital Lesson 6, Multiply Mania, students have two minutes to answer problems for multiples of six and seven. Problems are not always presented sequentially. For example, students may solve a string of problems 6 x 6, 6 x 9, 6 x 6, 6 x 8, 6 x 5, 6 x 9, 6 x 8, etc. During Teacher-Led Instruction students practice multiplying by 6 and counting forward and backwards from different starting points, for example, from 30 to 90.
• Conceptual understanding is embedded into every lesson. In Mission 3, Teacher-Led Instruction, Lesson 6, students apply the distributive property to multiply using units of 6 and 7. Students rewrite expressions into known facts. For example, 7 x 6 is rewritten into 5 x 6 and 2 x 6 so that 7 x 6 = (5 x 6) + (2 x 6). Students use this understanding to rewrite the division problem 48 ÷ 6 as the sum of two quotients (30 ÷ 6) + (18 ÷ 6).
• Application problems are embedded into every lesson and often call for students to model their thinking and make connections to procedural skills. In Mission 3, Teacher-Led Instruction, Whole Group Word Problem, Lesson 6, students are presented with the problem: “Mabel cuts 9 pieces of ribbon for an art project. Each ribbon is 7 centimeters long. What is the total length of the pieces of ribbon that Mabel cuts?” This problem builds on the procedural fluencies working with multiples of six and contextualizes the concept development on the distributive property.

The instructional materials reviewed for Zearn Grade 3 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. The students’ materials in the Teacher-Led 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:

• In Mission 1, Teacher-Led Instruction, Whole Group Word Problems, Lesson 8, students are asked, “Who is correct? Explain how you know, using models, numbers, and words.”
• Mission 3, Teacher-Led Instruction, Optional Problem Set Lesson 4, Question 5: “Julie counts by six to solve 6 x 7. She says the answer is 36. Is she right? Explain your answer.”
• Mission 5, Mid-Module Assessment Question 4: "Natalie says her drawing shows 3/2, and Kosmo says the picture shows 3/4. Students are supposed to “Show and explain how they could both be correct by choosing different wholes. Use words, pictures, and numbers.”
• Mission 7, End-of-Module Assessment, Question 3 Part b: Students are to determine if a student is correct that said two shapes having the same area must also have the same perimeter; and students should use their answer from Part a to explain why or why not.

Examples of opportunities to construct viable arguments:

• Mission 2, Teacher-Led Instruction, Lesson 17: Students talk to each other to determine which estimate is the closest. In Lesson 4, students explain which person in the problem finishes first and how they determined their answer.
• Mission 3, Teacher-Led Instruction, Lesson 2: Students explain their thinking using different representations while solving a problem about rainfall.
• Mission 5, Teacher-Led Instruction, Lesson 23: Students justify equivalent fractions to other students using number lines that have been divided into different intervals.

The Zearn Grade 3 materials meet the expectation 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, Lesson 1, Problem 3, teachers write an incorrect multiplication sentence on the board for students, telling the students to write an addition sentence and to “use your addition sentence as you talk to your partner about why you agree or disagree with my work.”
• In Mission 7, Teacher-Led Instruction, Lesson 2, Part 2, teachers are given the following questions to assist students in constructing viable arguments for their work: “How does your drawing represent the problem clearly?; How did your drawing help you decide on a way to solve?; Why does the equation that you used to model make sense with your drawing and with the problem?; and How do you know you answered the question?” In the same part of Lesson 2, teachers are given the following questions to assist students in analyzing the arguments of others: “I’m not sure what you mean. Can you say more about that?; Why did you decide ____?; What do you think about ____ instead?; Which other way did you try to draw the problem?”
• In Mission 7, Teacher-Led Instruction, Lesson 3, Problem 2, teachers are given the following prompts and questions to assist students in analyzing the work of a partner: “Study you partner’s work. Try to explain how your partner solved the problem.; Compare the strategies that you used with your partner’s strategies. How are they the same? How are they different?; What did your partner do well?; What suggestions do you have for your partner that might improve her work?; Why would your suggestions be an improvement?; and, What are the strengths of your own work? Why do some methods work better for you than others?”

The instructional materials reviewed for Zearn Grade 3 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 Lessons and then reinforced in the Whole Group Word Problems. Teachers, when applicable, model the vocabulary. For example, Mission 4, Teacher-Led Instruction, Whole Group Word Problems, Lesson 2 states, “This problem reviews the Lesson 1 concept that, although shapes look different, they may have the same area.”
• 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 1, Independent Digital Lesson 1, Math Chat introduces students to the terms, equation and multiplication and shows how these terms are used in many examples.
• Students are expected to use correct mathematics vocabulary as they Read, Draw, and Write for Word Problems. For example, in Mission 1, Teacher-Led Instruction, Whole Group Word Problems, Lesson 1, students must use correct terminology and representations as they determine how many total students are in the third grade.