4th Grade - Gateway 2

The instructional materials reviewed for Zearn Grade 4 meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific cluster headings, such as 4.NBT.A, 4.NBT.B, 4.NF.A, 4.NF.B, 4.NF.C, and 4.MD.C.

Place value concepts are expanded in Grade 4 to include fractions and decimals (4.NF.C), to generalize place value understanding for multi-digit numbers (4.NBT.A), and to use place value understanding and the properties of arithmetic to perform multi-digit arithmetic (4.NBT.B).

• In Mission 1, Teacher-Led Instruction, Small Group Lesson 2, students use a millions place value chart to understand the relationship of a digit in one place representing ten times what it represents to its right.
• Mission 3, Topic B develops conceptual models for multiplication by 10, 100 and 1,000.
• In Mission 6, Decimal Fractions, students extend place value understandings to numbers between 0 and 1, expanding the place value chart to tenths and hundredths and using their understanding of decimal place value to compare decimals. These concepts are further developed to addition with decimals and to understand money amounts as decimal values.

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.

• Missions 5 and 6 focus on clusters 4.NF.A and 4.NF.C. Tape diagrams, number bonds, concrete models, and number lines are used as models to demonstrate understanding of fractions. In Mission 5, Teacher-Led Instruction, Lesson 1, students use paper strips to decompose the whole into equal parts so as to demonstrate addition of fractions. The assessments for these missions include opportunities for students to shade, draw, or explain to justify their answers.
• Throughout the Grade 4 Missions and Lessons, students are frequently asked to draw and make conclusions based on their drawings.
• Assessments include opportunities for students to draw pictures, make models, and use place value. For example, the Mission 3, Teacher-Led Instruction Mid-Module Assessment Question 1 asks students to show understanding when multiplying and dividing multi-digit numbers by drawing pictures and using place value.

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 4 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 1, Teacher-Led Instruction Whole Group Fluency Lesson 12 (4.NBT.4), students practice rounding to the nearest ten thousand, thousand, hundred and ten place. They practice the standard addition algorithm with problems written vertically or horizontally or solved mentally. For example: 417 + 232 =_____. They repeat the process for finding sums for multi-digit numbers up to 1,000,000. For example: 23,944 + 6,056 + 159,368.
• In Mission 1, Independent Digital Lesson 12 Sum Sense, students practice solving word problems using the standard algorithm. During Blast students solve two-digit and one-digit addition problems (4.NBT.4).

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 4 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 5, Equivalent Fractions (4.NF.A) 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 situations of equivalent fractions. For example, in Teacher-Led Instruction, Whole Group Word Problems Topic B: Fraction Equivalence Using Multiplication and Division, “Students begin to generalize their work with fraction equivalence.” The topic includes four lessons:

• Lesson 7: “Model an equivalent fraction for 4/7 using an area model.” The teacher note states, “This application problem reviews Lesson 6 and leads into today’s lesson as students find equivalent fractions using multiplication.”
• Lesson 8: “Saisha gives some of her chocolate bar, pictured below, to her younger brother Lucas. He says, 'Thanks for 3/12 of the bar.' Shaisha responds, 'No. I gave you ¼ of the bar.' Explain why both Lucas and Saisha are correct.” The teacher note states, “...Revisit this problem in the Student Debrief (of the Concept Development part of the lesson) by asking students to write the remaining portion as equivalent fractions.”
• Lesson 9: “What fraction of a foot is 1 inch? What fraction of a foot is 3 inches? (Hint: 12 inches = 1 foot.) Draw a tape diagram to model your work.” The teacher note states, “Students are asked to think about fractions within a context, such as measurement, that will be useful in upcoming word problems.”
• Lesson 10: "Nuri spent 9/12 of his money on a book and the rest of his money on a pencil. a.) Express how much of his money he spent on pencils in fourths. b.) Nuri started with $1. How much did he spend on the pencil? The teacher note states, “This Application Problem connects Topic A and Lesson 9 by finding the other fractional part of the whole and expressing equivalent fractions.”

Throughout Grade 4 students deepen their understanding of multiplication and division as they apply concepts to their developing understanding of fraction equivalence. 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 4 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 5, Teacher-Led Instruction, Whole Group Fluency Lesson 2 builds on understandings from Grade 3 to help students recognize unit fractions. In Independent Digital Learning Blast, students practice division facts without remainders.
• Conceptual understanding is embedded into every Lesson. In Mission 5, Teacher-Led Instruction, Lesson 5, students use area models to show fraction equivalence. During Independent Digital Practice Tower of Power Lesson 5 students use the area model to show equivalence between ⅓ and 2/6. They complete equivalence sentences and addition sentences to show how the sum of two unit fractions in sixths equals one unit fraction in thirds.
• Application problems are embedded into every Lesson and often call for students to model their thinking and make connections to procedural skills. Mission 5, Teacher-Led Instruction, Whole Group Word Problems, Lesson 5 states: “A loaf of bread was cut into six equal slices. Each of the six slices was cut in half to make thinner slices for sandwiches. Mr. Beach used 4 slices. His daughter said, ‘Wow! You used 2/6 of the loaf!’ His son said, ‘No. He used 4/12.’ Work with a partner to explain who is correct using a tape diagram.”

The instructional materials reviewed for Zearn Grade 4 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:

• Mission 3, Teacher-Led Instruction Optional Problem Set, Lesson 22, Question 3: “Bryan says all prime numbers are odd numbers. List all of the prime numbers less than 20 in numerical order. Use your list to show that Bryan’s claim is false.”
• Mission 5, Teacher-Led Instruction Lesson 18: Students analyze the work of another student when adding or subtracting three fractions with the same denominator. Students offer different methods for completing the problems.
• Mission 7, End-of-Module Assessment, Question 6 Part d: “Jacob says that he can find the number of inches in 15 yards by tripling the number of inches in 5 yards. Does his strategy work? Why or why not?”

Examples of opportunities to construct viable arguments:

• In Mission 2, End-of-Module Assessment, Question 4 Part c, students explain their thinking as they determine a person’s weight in grams, before the person started training for a half marathon.
• In Mission 3, Teacher-Led Instruction, Optional Homework, Lesson 2, Question 4 Part f, students explain how to construct an argument, using words, pictures, or numbers when comparing how the perimeter changed with how the area changed (between two rugs).
• In Mission 4, Mid-Module Assessment, Question 3, students create a picture from 6 prompts using a protractor and straightedge, and they construct an argument for which lines are parallel in their drawing.
• In Mission 7, Teacher-Led Instruction, Optional Problem Set, Lesson 3, Question 5, students construct an argument for how they converted 23 hours and 5 minutes to minutes.

The Zearn Grade 4 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 Small Group 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 4, Problem 4, teachers write an incorrect number on the board based on a number written in expanded form, and after students write the number in a place-value chart, the teacher directs the students to compare their number with hers.
• In Mission 7, Teacher-Led Instruction, Small Group, Lesson 5, Problem 2, teachers give a form for critiquing work to student pairs and tell the students, “Work with a partner to complete the Problem Set. When you are finished solving and creating a word problem to go along with each diagram, turn to your partner and share. Use the peer share and critique form to take notes about your work and your partner’s work.”
• In Mission 7, Teacher-Led Instruction, Small Group, Lesson 8, Problem 1, each of three students presents a different way to solve a problem involving addition of weights in pounds and ounces, and the teacher directs other students to question the presenting students in order to understand each of the solution strategies.

The instructional materials reviewed for Zearn Grade 4 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 5 states, “The Application Problem reviews solving for an unknown side length (Module 4) and metric conversions (Module 2). Division of decimals is a Grade 5 standard, so instead, students might convert to centimeters (as in Solution A), use their fraction knowledge to decompose 48 hundredths into 4 equal parts (as in Solution B), or simply think in unit form, i.e., 48 hundredths ÷ 4 = 12 hundredths.”
• 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 4, Independent Digital, Lesson 3 Math Chat introduces students to the term, perpendicular lines, and has students complete several examples identifying lines or line segments that are perpendicular and ones that are not perpendicular.
• Students are expected to use correct mathematics vocabulary as they Read, Draw, and Write for Whole Group Word Problems. For example, in Mission 1, Teacher-Led Instruction, Whole Group Word Problems, Lesson 5, students must use correct terminology and representations as they create numbers that meet given criteria using a place-value chart.