The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

Lessons are designed to support students to explore and develop conceptual understanding of grade-level mathematics. For example, students develop conceptual understanding:

• In Lesson 8 Session 3, Develop, students model problems using both pictures and arrays to determine the factors of a number. The teacher’s guide prompts teachers to ask students the following questions, “Where does your model show the total number of cars in each arrangement of cars? The number of cars in each same-sized row? The number of rows?” (4.OA.4)
• In Lesson 11, Session 2, Develop, students are shown models of base ten blocks and partial products to multiply. Teachers are instructed, “Compare and connect the different representations and have students identify how they are related.” Teachers ask, “Where does your model show the number of hundreds, tens and ones in 254? the multiplication of each place value by 3? the product?” These questions help students build conceptual understanding of multiplication by one-digit numbers (4.NBT.5)
• In Lesson 12, Session 2, Develop, Try It, students develop conceptual understanding of multiplying two two-digit numbers, using strategies based on place value and the properties of operations (4.NBT.5), as students are presented with the problem “16 x 28” and shown solutions using an area model and partial products.  Students are then asked to reflect as to how the partial products method relates to the area model method. 
• In Lesson 18, Session 2, Develop, Try It states, “A grasshopper weighs $$\frac {2}{100}$$ of an ounce. A beetle weighs $$\frac {8}{10}$$ of an ounce. Which weighs more?” Students are prompted to use a math toolkit in the margin which lists, “number lines, hundredths grids, tenth grids, index cards, and fraction models.” (4.NF.2)

In the Student Worktext and during Interactive Practice, students have opportunities to independently demonstrate conceptual understanding. For example:

• In Unit 1, Math in Action, Session 1, students work independently to develop a conceptual understanding of place value and expanded form as they solve “Yearly Blog Visits.”  Students analyze six months of blog visitors with totals represented in expanded form, standard form, and written form. The students are asked, “How many visitors should Max expect to get on his blog site in one year?”  Students complete a graphic organizer based on the presented data with the following prompts, “I know..., I need to find..., I can find the solution by....” Students then reflect upon their solution “I know my solution works...” (4.NBT.2 & 4.NBT.3) 
• In Lesson 15, Session 2, Develop, Problem 2, students use partial quotients to solve “$$4,507\div4$$.” (4.NBT.6)
• In Lesson 27, Student Worktext, Session 2, Develop, Practice Comparing Decimals in Hundredths, Problem 1 states, “Shade and label the models to show .33 and .35. Then explain how the models show which decimal is less.” (4.NF.7)
• In Interactive Practice, Understand Place Value, students reason about numbers and their relationships to determine the place value of a digit. (4.NBT.1)
• In Interactive Practice, Divide Three Digit Numbers, students use tape diagrams and break apart numbers to understand how to divide three-digit numbers. (4.NBT.6)

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. 

The materials include problems and questions, interactive games, and math center activities that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. 

• In Lesson 4, Session 2, Develop states, “Sam adds 6,152 and 379 and gets a sum of 9,942.” They must also explain why Sam’s addition is incorrect and find the correct sum of 6,152 + 379.  Students develop procedural skills by connecting place value methods to use of the standard algorithm to add multi-digit whole numbers. 
• In Lesson 11, Interactive Tutorial, students use partial products to multiply three-digit numbers by a one-digit number.  For example, 7 X 328 = 7 X 300 + 7 X 20 + 7 X 8 = 2100 + 140 + 56 = 2296. They look at patterns to determine the product of a one-digit and 4-digit number. Students practice multiple questions with patterns like this throughout the tutorial.  

The instructional materials include Learning Games, interactive games to help build procedural skill and fluency, available in both English and Spanish, that can be accessed through i-Ready Reports. For example, Grade 4 students can play the game “Hungry Fish” which allows them to strengthen skills in the area of computation (4.NBT.4, 4.NBT.5). In this game, students are tasked with solving addition, subtraction, multiplication, or division problems using bubbles with parts of an equation and a fish with a corresponding solution. 

4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm) requires students to develop grade level fluency. This standard is addressed in several lessons in the Student Worktext, including:

• In Lesson 4, Session 4, Refine, Apply It, Problem 1 states, “The population of Turtle Valley is 407,989. The population of Art Creek is 86,966. What is the total population of the two cities? Show your work.” 
• In Lesson 5, Session 4, Refine, Apply It, Problem 1 states, “What is the difference of 484,392 and 53,674? Show your work.” Students are expected to use the standard algorithm to solve the problem.

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Within each lesson, there are Fluency and Skills Practice pages that children complete on their own. In addition, there are Learning Games and Math Center Activities that engage students with fluency practice. Examples of when students get opportunities to independently demonstrate procedural skill and fluency are:

• In Lesson 5, Session 2, Develop, Practice Using Strategies to Subtract, Problem 2 asks students to “Find the difference. 16,407 - 9,524.”  Students develop the procedural skill of subtraction of multi-digit whole numbers by connecting place value concepts to the standard algorithm. (4.NBT.4)
• In Lesson 14, Fluency and Skills Practice, Dividing with Arrays and Area Models, Problem 8 states, “488 ÷ = __.” Problem 12 states, “366 ÷ 6 = __.”
• In Lesson 32, Fluency and Skills Practice, students practice adding and subtracting angles to make an angle of 180 degrees. (4.MD.7)
• Math Center Activities are provided for students to work together in partnerships to develop procedural skill and fluency. In Lesson 4, Adding Whole Numbers, students develop procedural skills in adding and subtracting multi-digit whole numbers using the standard algorithm (4.NBT.4) in a game format with a partner.  In the game Find Sums, students use a recording sheet containing two four-digit numbers, two five-digit numbers, and two six-digit numbers and a game board to solve addition problems such as, “4,376 + 1,337.” When students select a problem to solve and accurately solve it, they get to mark off the matching sum on the game board. When students have three numbers marked in a row, they win the game.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

Examples of opportunities for students to engage in routine application of mathematical skills independently and to demonstrate the use of mathematics flexibly in a variety of contexts in Classroom Resources include: 

• In Lesson 10, Session 3, Develop, Apply It, Problem 8 states, “Cadence shops for hiking gear. She picks out a $95 tent, a pair of boots for $54, and a $38 backpack. She has $200 to spend on gear. Write and solve an equation to find out if she has enough money for the hiking gear. Estimate to check that your answer is reasonable. Show your work.”
• In Lesson 16, Fluency and Skill Practice, Problem 6 states, “Nora uses 80 inches of string to make a border for a rectangular painting. She wants the painting to have a width of 15 inches. What will be the length?”
• In Lesson 20, Session 5, Refine, Apply It states, “Emily eats $$\frac{1}{6}$$ of a bag of carrots. Nick eats $$\frac{2}{6}$$ of the same bag of carrots. What fraction of the bag of carrots do Emily and Nick eat altogether?”
• In Lesson 7, Math Center Activity, Multiplication and Division in Word Problems provides students with the opportunity to multiply or divide to solve word problems involving multiplicative comparison (4.OA.2). The materials state, “Flora buys 6 apples and 24 oranges. How many times as many oranges as apples does Flora buy?”

The instructional materials include multiple opportunities for students to engage in non- routine application of mathematical skills and knowledge of the grade level.  For example, in Classroom Resources:

• In Unit 2, Math in Action states, “The zoo is planning to build a new area for birds. The zoo is going to use recycled materials. There will be three different-size rectangular cages as shown below: Small cage: floor area of 12 square feet, Medium cage: floor area of 24 square feet, Large cage: floor area of 36 square feet. Beau needs to find a possible length and width for the rectangular floor of each size cage. What is a possible length, width, and perimeter for each cage’s floor?” 
• In Unit 4, Math in Action, Session 2 states, “Luna is designing a picture frame made out of craft sticks. Below are her instructions: Paint 6 craft sticks. Each stick is $$\frac{3}{4}$$ of an inch wide and $$5\frac{3}{4}$$ inches long. Glue the craft sticks side-by-side on a piece of cardboard. Glue a photograph $$2\frac{1}{4}$$ inches wide and $$2\frac{1}{4}$$ inches tall on the frame. Leave a space at least $$2\frac{2}{4}$$ inches wide of the right of the photo. You can put your decorations here. There needs to be at least $$\frac{2}{4}$$ of an inch of space above and below the photo. Explain if Luna’s plan works.”
• In Unit 3, End of Unit, Unit Review, Performance Task states, “Zander expects 225 guests to be at the party. He plans on having 8 items in each goodie bag. Use the list of items below to put together an order for Zander. Explain how you know that your order has enough items.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The instructional materials address specific aspects of rigor, and the materials integrate aspects of rigor.

 All three aspects of rigor are present independently throughout the program materials. For example:

• In Lesson 1, Session 2, Develop, Connect It, students develop conceptual understanding of place value. Problem 5 states, “What do the expanded form and a place-value chart tell you about a number such as 25,049?  How are they alike and different?” 
• In Lesson 4, Session 4,Refine, students add multi-digit numbers using the standard algorithm. Problem 8 states, “Find the sum of 6,618 and 132,501.  Then estimate to check that your answer is reasonable. Show your work.” 
• In Lesson 10, Session 4, Refine, students solve multi step word problems posed with whole numbers.  Problem 2 states, “Taylor earns $5 each time she walks her neighbor’s dog. She has already earned $25.  Write and solve an equation to find out how many more times Taylor needs to walk the dog to earn enough to buy a bike that costs $83.  Check the reasonableness of your answer. Show your work.” 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example: 

• In the online game platform there are several opportunities for students to  engage students in all three aspects of rigor: conceptual understanding, fluency and application. The game “Pizza” focuses on mental math and economic applications. Students use fluency and conceptual understanding to solve application problems related to costs associated with a pizza restaurant. Students set prices, compare vendors to find the best  prices for pizza ingredients, and calculate the cost of customers orders. 
• In Lesson 21, Session 3, Develop, students use models to connect written procedures for subtracting mixed numbers in word problems. Problem 7 states, “Monica rides her bike $$3\frac{1}{4}$$ miles on Monday.  She rides $$2\frac{2}{4}$$ miles on Tuesday. How much farther does Monica ride on Monday than on Tuesday? Show your work.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. Overall, the materials attend to aspects of the mathematical practices (MPs) during different lessons throughout the grade, so when taken as a whole, the instructional materials attend to the full meaning of each MP.

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Program Implementation, Standards for Mathematics in Every Lesson, includes information on how the Standards for Mathematical Practice (MPs) are addressed within each lesson, noting that key components of lessons are designed to engage students with the MPs. Deepen Understanding provides guidance for teachers on each MP within lessons sessions. Discourse Questions are included, as well as prompts for Structure and Reasoning specifically related to MP7 (Look for and make use of structure) and MP8 (Look for and express regularity in repeated reasoning). In addition, the Try-Discuss-Connect Instructional Routines all identify the MPs. 

The instructional materials attend to the full intent of all eight Mathematical Practices. For example:

MP 1: Make sense and persevere in solving problems.

• In Lesson 8, Session 3, Develop, Try It states, “Alfred is arranging 40 model cars into rows. He wants to put the same number of cars in each row. Find all the ways he can arrange the cars.” Teacher guidance states, “Make sense of the Problem: To support students in making sense of the problem, have them show that they recognize the need to find different ways that Alfred can arrange model cars in rows that have the same number of cars in each row.” Student Worktext, Discuss It states, “Ask your partner: How did you get started? Tell your partner: I started by...”
• In Lesson 13, Session 2, Develop, Try It states, “Wanda is shopping for a pet carrier for her cat. One small carrier can hold 240 ounces. Her cat weighs 12 pounds. Can the carrier hold her cat?” Teacher guidance includes, “To support students in making sense of the problem, have them identify that the Customary Units of Weight table shows how pounds and ounces are related. Students may want to use the Math Reference Sheet as a reference for equivalent measurements of weight.”

MP 2: Reason abstractly and quantitatively.

• In Student Worktext, Lesson 14, Session 2, Develop, Try It states, “What is 136 ÷ 4?” Discuss It states, “Ask your partner: Why did you choose that strategy? Tell your partner: I do not understand how…” 
• In Unit 2, End of Unit, Unit Review, Performance Task states, “Melanie is in charge of setting up chairs for the Student Jazz Concert. She has 96 chairs to arrange in rows in a large classroom. She wants each row to have the same number of chairs. Find at least 6 different ways that Melanie can arrange the chairs. Choose one arrangement and explain why it does not make sense to use that arrangement. Then choose the arrangement that you think works best.”  Students reason quantitatively when determining factor pairs for the six arrays, then reason abstractly when determining which factor pairs are most suitable for arranging the chairs.

MP 4: Model with mathematics.

• In Unit 5, Math in Action, Session 2 states, “Bella is designing a wood mosaic piece. Angle Cuts: When Bella cuts pieces of wood for a project, she always saves the scraps. The scarps make good pieces for making mosaics. The cut pieces of different angle measurements. Bella sorts pieces by the angle measurements (angles are sorted into 30°, 45°, 50°, 60°, 80°, 90°, 100°, and 120°). Find a way Bella can put some pieces of wood together to make a 180° angle. Then measure and draw your angles to show that they make a straight line.” Reflect states, “Use a Model: What equations could you write to show the total measure of the angles you put together?”
• In Lesson 29, Session 2, Develop, Apply It, Problem 7 states, “Lulu has 10 feet of ribbon. She uses $$1\frac{1}{3}$$ feet of ribbon for a project. She uses the rest of the ribbon to make bows. She uses 8 inches of ribbon for each bow. How many bows does Lulu make?” Students are to “Use what you have just learned to solve these problems.” Students can use equations, bar drawings, and/or number lines. 

MP 5: Choose tools strategically. 

• In Student Worktext, Lesson 14, Session 3, Develop, Try It states, “There are 232 people waiting in line for an amusement park ride. Each car on the ride will be filled with 5 people. How many cars are needed to hold all the people waiting in line?” The Math Toolkit includes base-ten blocks, grid paper, and multiplication models for students to choose.
• In Student Worktext, Lesson 26, Session 1, Explore, Try It, the Math Toolkit includes base-ten blocks, play money, hundredths grids, and index cards. Students choose tools to solve “Max has 248 pennies. How many whole dollars does Max have? What fraction of a dollar is left over?”

MP 6: Attend to precision.

• In Lesson 16, Session 1, Explore, Support Partner Discussion, provides teachers with guidance for helping students attend to precision as they discuss, “Marissa uses 64 feet of fence to make a border around a rectangular flower garden. The length of the garden is 20 feet. What is the width of the garden? Encourage students to use the terms perimeter, width, and length as they discuss their solutions.”
• In Unit 4, Unit Review, Performance Task states, “Ciara is using the recipe below to make 6 dozen cupcakes for a family party. She needs to buy flour, milk, and vanilla. Ciara also needs to buy boxes to carry the cupcakes to the party. Each box holds one layer of cupcakes. Ciara has $25 to spend. Does she have enough money to buy everything she needs to make the cupcakes and bring them to the party? Explain how you know.” The recipe and the prices for each ingredient are given. Reflect states, “Why is it important to use labels for all of the amounts while you are solving this problem?”

MP 7: Look for and make use of structure. 

• In Lesson 11, Session 2, Develop states, “What is the product of 3 and 254?” In Deepen Understanding, Partial Products, MP7 Make use of structure states, “When discussing partial products, prompt students to see that 3 x 254 can be written as 3 x (200 + 50 + 4) using the expanded form of 254. Ask: What idea that you learned can you use to find 3 x 254? Listen for: You can multiply each addend, 200, 50, and 4, by 3.” In this example, students use the structure of the distributive property to multiply a three-digit number by a one-digit number.
• In Lesson 20, Session 2, Develop states, “Josie and Margo are painting a fence green. Josie starts at one end and paints 3/10 of the fence. Margo starts at the other end and paints 4/10 of it. What fraction of the fence do they paint altogether?” In Deepen Understanding, Number Line Model, MP7 Look for structure states, “When discussing the number line model, prompt students to consider how it could be used to demonstrate the commutative property. Generalize, Do you think this is true no matter what numbers you are adding? If you were using a number line to add 3 and 4, would it be true? Have students explain their reasoning. Listen for understanding that when adding whole numbers or fractions, the order of addends does not matter, the sum stays the same.”

MP 8: Look for and express regularity in repeated reasoning.

• In Lesson 8, Session 2, Develop states, “Leona has 5 cups of oats. She needs 2 cups of oats for one full batch of oatmeal muffins. Can she use all of her oats by making multiple full batches of muffins?” In Deepen Understanding, Number Line Model, MP8 Use repeated reasoning states, “When discussing the number line model, prompt students to consider how it could be used to find multiples of another number, such as 3. Ask: How can you find multiples of 3 on the number line? Listen for: Start at 0 and skip-count by threes. Ask: What multiples of 3 do you see on the number line? Listen for: The multiples of 3 on the number line are 3, 6, and 9. Ask: How can you tell which number is a multiple of both 2 and 3? Listen for: A number that has both a circle and a box around it is a multiple of 2 and 3, so 6 is a multiple of 2 and 3.” 
• In Lesson 26, Session 2, Develop, Model It states, “You can use a place-value chart to understand how to write hundredths or tenths as a decimal. The place-value chart shows the value of 0.60.” Deepen Understanding states, ”When discussing the place-value chart, prompt students to consider how it helps them read the decimal and identify the place value of each digit.” It provides teachers with the questions, “What is the least place value of the number shown in the chart? What is the connection between the least place value and how you read the number? Do you think that you can always read a number in the place-value chart by the place value of the last digit shown?”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Classroom Resources, the Student Worktext and the Math Journal provide students with opportunities to construct arguments and critique the reasoning of others. Evidence where students have opportunities to construct viable arguments includes: 

• In Lesson 7, Session 1, Try It, students solve a multiplication word problem and then justify their reasoning using the Discuss It prompt states, “Ask your partner: Do you agree with me? Why or why not? Tell your partner: I agree with you about… because…”
• In Lesson 17,  Student Worktext, Session 3, Develop, Problem 4 states, “Explain why $$\frac{3}{4}$$ is equivalent to $$\frac{9}{12}$$.”
• In Lesson 18, Session 4, Refine, a series of Pair/Share prompts in the margin of the materials help students construct viable arguments. The materials state, “How else could you solve this problem? Which strategy for comparing do you think works best with these fractions? How did you and your partner decide what strategy to use to solve the problem?”

Evidence where students have opportunities to analyze the mathematical arguments of others includes:

• In Lesson 33, Session 3, Develop, Discuss It states, “Ask your partner: Do you agree with me? Why or why not? Tell Your Partner: I agree with you about...because…” 
• In Lesson 4, Session 3, Develop, Problem 5, students solve, “There is a mistake in the addition problem shown.  Explain how the mistake was made. Then find the correct sum. 22,365 + 53,908 = 75,373"

Evidence where students constructing viable arguments and analyze the mathematical arguments of others includes:

Throughout the series there are dialogue boxes with the phrase, “Discuss It.” This dialogue box often encourages students to engage in discourse about the mathematics of the lesson. For example, in Lesson 7, Session 1, Explore, Try It, students solve the problem, “Hannah scored 3 goals last season. She scores 4 times as many goals this season. How many goals does Hannah score this season?” Discuss It, “Ask your partner: Do you agree with me? Why or Why not? Tell your partner: I agree with you because…”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Program Implementation, Resources include Discourse Cards which present questions and sentence starters to engage students in mathematical discourse, including the construction of arguments and analysis of others reasoning. For example, “Can you convince your partner or others that your answer makes sense? What do you think about what another student said? Does your partner’s strategy make sense? How is your solution method the same as or different from another student’s method?

In Classroom Resources, Lesson 0, Understanding the Try-Discuss-Connect Instructional Routine. The Discuss routine presents opportunities to use questions and sentence starters for students to share their thinking and critique each other’s reasoning. In addition, using Compare Strategies, students discuss how representations are the same, different, and related. For example, Lesson 0, Session 1, Discuss It, Compare Class Strategies states, “There are 236 third graders at Huron Elementary School. What is 236 rounded to the nearest hundred? Explain your reasoning.” Once students share strategies and reasoning, the teacher asks “How are they the same? How are they different? How are they connected?” These instructional routines are present in every lesson.

Evidence where the instructional materials support teachers to engage students in constructing viable arguments includes:

• In Lesson 14, Session 3, Discuss It states, “There are 232 people waiting in line for an amusement park ride. Each car on the ride will be filled with 5 people.  How many cars are needed to hold all the people waiting in line?” Teachers are prompted to ask such questions as, “Can you explain why you did it that way?”
• In Unit 5, End of Unit, Unit Review, Performance Task, Reflect, has students reflect on their interpretations of logos they drew on grid paper matching a description. The materials state, “Argue and Critique: Suppose you ask your uncle to draw a log matching your description. Do you think his drawing will match what you had in mind? Explain your answer.” Teacher guidance about this section says, “Argue and Critique: Students should explain that there can be more than one figure that matches a given description.”

Evidence where the instructional materials support teachers to engage students in analyzing the arguments of others includes:

• In Lesson 11, Session 1, Explore, Discuss It states, "What is the product of 3 and 57? Ask: How do (student name)'s and (student name)'s models represent 57 three times.
• In Lesson 25, Session 2 states, “Carmen has $$\frac{4}{10}$$ of a dollar. Troy has $$\frac{50}{100}$$ of a dollar. Together, what fraction of a dollar do they have?” In Deepen Understanding, Models of Tenths and Hundredths, SMP3 Construct arguments and critique reasoning, states “Provide an opportunity for students to practice presenting their reasoning and critiquing the reasoning of others.  Have several students present their reasons for representing Carmen and Troy’s money as they did. Ask: Why is your model or strategy a good way to show adding fractions with denominators for 10 and 100? Listen for: Responses should include specific advantages or strengths of students’ chosen model or strategy. Ask: What questions do you have about [student name]’s model or strategy? What do you think is helpful about the model or strategy [student name] used? What might be confusing or unclear in [student name]’s work using that model or strategy? Listen for: Responses should include clarifying questions and support identification of strengths and/or weaknesses of students’ models or strategies.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations that materials explicitly attend to the specialized language of mathematics.

In i-Ready, Teach and Assess, Ready Classroom Mathematics, there are several resources to support teachers and students to use the specialized language of mathematics. In Program Implementation, the Academic Vocabulary Glossary identifies the vocabulary, provides a definition, and uses the word in a sample sentence organized by unit. For example, “Clarify, When you clarify a problem, you make it easier to understand”. 

In Classroom Resources, Lesson Overview, Lesson Vocabulary identifies whether there is new vocabulary or review, and key terms used in the lesson. For example in Lesson 18, Session 2, vocabulary includes: “common denominator a number that is a common multiple of the denominators of two or more factors; denominator the number below the line in a fraction that tells the number of equal parts in the whole; numerator the number above the line in a fraction that tells the number of equal parts that are being described.” 

Throughout lessons in Ready Classroom Mathematics, students are provided with Graphic organizers that assist with mathematical language to ensure students are using precise vocabulary. There are five different graphic organizers used throughout the lessons that allows students to organize learning concepts and vocabulary through definitions, illustrations, examples, etc. For example, Students Worktext, Lesson 14, Session 1 states, “Think about what you know about unknowns in equations. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Unknowns: In My Own Words, My Illustrations, Examples, Non-Examples.”

Build Your Vocabulary is provided at the beginning of each unit to support students in learning and using precise language and terminology. For example in, Unit 2, Beginning of Unit, Build Your Vocabulary, there are review vocabulary words, “array, equation, grouping, and factor.” The teacher’s guide provides suggestions for teachers to use with students to build math vocabulary using this page. It states “Display, point to, and read each review word aloud. Have students repeat chorally.” The page has diagrams for each word. The teacher’s guide states, “Have students work independently to label each word using the Review words. Pair students. Ask students to share their answers with their partners and explain how they decided which label to use. Observe and listen to students’ discussions to assess their understanding.”

In the Lesson Overview, Language Objectives are included for each lesson. For example, Lesson 8, Lesson Overview, Language Objectives, includes “Orally define and use in discussion the key mathematical terms factor, factor pair, multiple, composite number and prime number.”

In the End of Unit, Vocabulary, Vocabulary Cards are available. The materials state, “The purpose of the vocabulary cards is to reinforce students’ understanding of the new vocabulary words in the unit as well as to provide a place for students to record any other math words and definitions that would be helpful for them in their understanding of unit concepts. Students may find it useful to draw or write examples on the vocabulary cards as they encounter the terms during the unit. you may want to make a copy of the cards and display them in a word wall in the classroom to further support students’ learning.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for distinguishing between problems and exercises. 

There are several practice pages provided with each lesson. After receiving guidance from the teacher in the Try It section of each lesson, students can demonstrate their understanding in a variety of ways. Students solve problems to learn new mathematics in the Explore sessions of each lesson. These ideas are further developed in Develop sessions, where students solve problems in the Try It and Connect It sections. In the Refine session, students complete exercises where they apply their learning. For example:

• In Grade 4, Lesson 23, Session 1, Explore, students use models to understand multiplication of fractions and whole numbers. In Session 2, Develop, students solve fraction multiplication by whole number problems using area models and number lines. In Session 3, Refine, students apply their learning in the Apply It session, where they solve word problems.  

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for designing assignments not being haphazard and exercises given in intentional sequences. 

The sequence of lessons in each topic is designed to move from concrete and pictorial representation toward abstract work with numbers. Each unit has a Unit Flow and Progressions Video that highlights the work of the unit and how it fits in the progression of mathematics across grade levels. 

Each unit also has a Learning Progression Chart that shows which lessons are building upon and which lessons students are preparing for within the unit. Each lesson has learning progression information which highlights work done in previous grade levels, as well as the work to be done in this lesson and subsequent lessons.

Each lesson has a consistent structure that builds towards independence. For example,

• In Grade 4, Lesson 14, Session 2, Develop, the teacher is prompted to connect to prior knowledge of the relationship between multiplication and division. The teacher is prompted to help students make sense of the problem, “What is 136 ÷4?” Teachers are provided with possible prompts to support partner discussion such as, “How did you get started? What tools or models did you use?” During Connect It, the teacher leads a discussion to help students connect various models used. Finally, during Apply It and the Exit Ticket students practice and independently show their understanding.

Lessons within units build upon each other. For example: 

• In Grade 4, Unit 2 includes the following lessons and topics: Lesson 6: Understand Multiplication as a Comparison, Lesson 7: Multiplication and Division in Word Problems, Lesson 8: Multiples and Factors, Lesson 9: Number and Shape Patterns, and Lesson 10: Model and Solve Multi-Step Problems.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for having a variety in what students are asked to produce. 

Students are expected to respond to problems in a variety of ways, including produce answers, use models, drawings, and equations to support their explanations. Students are asked to justify their solutions with a partner and participate in discussions with Discuss It and Pair/Share prompts. Students respond to different problem types in the Refine section of the lessons, including short answer explanations, multiple choice, fill in the blank, and drawings. For example:

• In Grade 4, Lesson 22, Session 4, Refine, students respond to problems in multiple ways when subtracting fractions in line plots. There are problems where students use tape diagrams, number lines, solve word problems, provide solutions with explanations, and solve multiple choice items. 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for manipulatives being faithful representations of the mathematical objects they represent. 

There are a variety of manipulatives in the Math Tool Kit available during lessons, and students are introduced to these in appropriate contexts for the concept being developed.

• In Grade 4, Lesson 11, Session 2, Develop, students use base ten blocks, grid paper, index cards, sticky notes, and number lines to multiply three-digit numbers by one-digit numbers. 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing teachers with questions that are designed to elicit students’ mathematical understanding and thinking. 

• In Grade 4, Lesson 15, Session 2, Develop, questions are posed for the teacher to ask students.  The questions are in italics and easy for the teacher to see. In Discuss It the following questions are provided: “How would you explain the strategy you used to solve the problem? Do you think your answer makes sense?”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meets expectations for including ample notes and useful annotations and suggestions on how to present the content in the student edition. 

The Program Implementation tab includes a “Digital Math Tools - Support Videos” section. This section includes support videos for counters and connecting cubes, base ten introduction, base ten: add and subtract, number line, multiplication models, perimeter and area, fraction models: add and subtract, and fraction models: compare and multiply.

In Classroom Resources, guidance for teachers supports the delivery of the content, as well as information on student responses for each section of the lesson. For example: 

• In Grade 4, Lesson 16, Session 2, Develop, the teacher notes include guidance for developing language. The materials state, “Develop Language Why: Support student understanding of the word edge as used in Apply It problem 7. How: Ask students to define or give examples of the word edge. Explain that the word edge is used informally to mean the end of an object. For example, students may sit on the edge of a rug or where the rug ends. In problem 7, edge refers to the side of a poster. Show students a poster and point to its edge.” This session also includes notes for common misconceptions, deepen understanding, and a hands-on activity if students are struggling. 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for containing a teacher’s edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject as necessary. 

In Classroom Resources at the beginning of each unit, Learning Progressions, Math Background, and Unit Flow and Progression Videos provide information for teachers on mathematics and models. For example:

• In Grade 4, Unit 4, Beginning of Unit, the Unit Flow & Progression video builds teacher understanding of how equivalent fractions in measurement and area are related to addition and subtraction from Grade 3, and leads to comparing fractions with unlike denominators, and how addition with mixed numbers can build on these underlying concepts. 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for containing a teacher’s edition that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for Kindergarten through Grade 12.  

The Ready Classroom Mathematics materials include a “Learning Progression,” section in the teacher edition for each lesson that describes how the grade-level appropriate standard is developed in previous grades as well as how it will extend in the next grade. For example:

• In Grade 4, Lesson 31, Full Lesson, Standards 4.MD.5 and 4.MD.6 are listed and defined.  Under the Learning Progression section on the same page, which describes the progression students will learn about angles, it starts with in the previous lesson and moves to in this lesson and in the next lesson.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing strategies for gathering information about students’ prior knowledge within and across grade levels. 

The materials in Ready Classroom Mathematics list prerequisite skills for each lesson in the Lesson overview. Prerequisite Lessons are provided with each lesson to review concepts or to provide students with instruction in areas that may be gaps in their learning. For example:

• In Grade 4, Lesson 20, the Prerequisite Lesson is: Grade 3, Lesson 20. 

In addition to identifying prerequisite skills for each lesson, Explore, at the beginning of each lesson, connects students’ prior knowledge with the content of the lesson. In some Warm-up questions there is a tag “Prerequisite” noting that the question is assessing prior knowledge.  

The instructional materials include adaptive Diagnostic Assessments with Prerequisite  Reports found in i-Ready, Reports.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions. 

The Ready Classroom Mathematics materials provide strategies for teachers to identify and address common student errors and misconceptions. For example,

• In Grade 4, Lesson 11, Session 2, Develop, the Common Misconception states, “Look for students who multiply each digit in 254 by 3 rather than multiplying the value of each digit by 3.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. 

Over the course of a lesson, which includes multiple sessions, materials engage students with multiple activities to review, practice, and independently demonstrate the grade-level mathematical concepts and skills. 

Feedback is provided to students as they progress through the Sessions. Frequent feedback opportunities to address skills and concepts are provided in the Classroom Resources tab, within each lesson and its sessions. The Reteach - Tools for Instruction within each lesson provides teachers with sample errors and remediation strategies to address those errors. In addition, throughout the lesson sessions, potential misconceptions are highlighted with guidance for teachers to provide feedback and new opportunities for practice. For example:

• In Grade 4, Lesson 14, Reteach Tools for Instruction, Divide Three-Digit by One-Digit Numbers, Check for Understanding states, “If you observe the student has trouble using multiplication facts to determine the missing factor, the student may have trouble recalling multiplication facts.  Then try providing a multiplication fact table and instructing the student on how to use it.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for offering ongoing formative and summative assessments that clearly denote which standard is being emphasized.

• Standards are clearly noted within the assessments found at the end of each lesson. Standards are provided alongside the questions as well as the Depth of Knowledge level. Unit Assessment Correlations are available with the question number, DOK, Standard, and Lesson. 
• A Mid-Unit Assessment is provided for longer units. Mid-Unit Assessments also provide standards correlations for each item. These can be found in the Classroom Resources tab, End of Unit, Assess, Lesson Quizzes, and Unit Assessments. 
• Formative assessments are also available:
• Lesson quizzes, exit tickets, and quick checks are provided for most lessons. These quizzes assess the specific standard(s) being taught in the lesson.
• Within the sessions are Check For Understanding supports that include a statement as to why the check is being done. The why relates back to the standard being taught in the activities for that session.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for including aligned rubrics and scoring guidelines to support teachers in interpreting student performance and sometimes provide for follow-up instruction. 

Rubrics are provided for Mid-Unit Assessments, Unit Assessments, Unit Performance Tasks, Lesson Quizzes, and Math in Action. (grade 2 and above) The rubrics and scoring guidelines support teachers to interpret student performance. Assessment answer keys are provided alongside the questions along with the DOK level for the item. For multiple choice answers, the correct answer and explanations for incorrect choices are given, including the most common misconceptions.  

Reports from i-Ready diagnostic assessments and comprehension checks assist teachers in providing follow-up instruction for misconceptions and mathematical errors. The electronic reports also include links to assist in reteaching items missed. Within lessons, rubrics and scoring guidelines provide guidance for teachers to follow-up and, throughout Ready Classroom Mathematics, there is guidance for teachers on behaviors to look for, error alerts, and Common Misconceptions. 

• Unit Assessments include a Unit Assessment Teacher Guide that provides instructors with solutions, points possible, the exact standard covered, and the depth of knowledge (DOK) level for each item.
• The Scoring Guide for the problems in the scoring table include: DOK, points of scoring, standard addressed, and lesson assessed by each problem.
• The Scoring Rubrics includes points and expectations for short response, multiple select, choice matrix, extended response, and fill-in-the-blank items.
• Assessment Practice provides some support for follow-up by identifying standards that need further study and reinforcement. Assessment Practice Answer Key and Correlations identify the standard which each question has been designed to evaluate. 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners. 

The sessions within a lesson follow the sequence of Explore, Develop, and Refine. Within a session, there is a sequence of Start, followed by Try It, Model It, or Apply It; Discuss; Connect; and Close.

• Start is designed to build fluency or connect to prior knowledge. 
• Try It is designed to give students an opportunity to explore the concept on their own.
• Model It is designed to allow students the opportunity to explore the concept through manipulatives or drawings. 
• Apply It is designed to give students an opportunity to practice the skill on their own. 
• Discuss is designed to allow students to talk to other students about the concept and compare what they did with each other. 
• Connect It is designed to help students connect the concept to real-life. 
• Close is designed to solidify the learning for the day and to check for understanding through the use of an exit ticket. 

Each lesson includes a Differentiated Instruction Teacher Toolbox that includes Reteach (Tools for Instruction), Reinforce (Math Center Activities), and Extend (Enrichment Activities). For example, the Lesson 1 Overview states: 

• Reteach- “Children who require additional support for prerequisite or on-level skills will benefit from activities that provide targeted skills instruction.”
• Reinforce- “Children who require additional practice to reinforce concepts and skills and deepen understanding will benefit from small group collaborative games and activities.” This is available in three versions: on-level, below-level, and above-level.
• Extend- “Children who have achieved proficiency with concepts and skills are ready for additional challenges will benefit from group collaborative games and activities to extend understanding.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing teachers with strategies to meet the needs of a range of learners. 

Each lesson provides opportunity for differentiation in small group lessons, interactive digital tutorials, support for teachers to address common misconceptions at different points throughout lessons. For example:

• The Differentiated Instruction opportunities provide support to reteach students who need additional support, reinforce learning for students who need additional practice, and extend learning for students who have achieved proficiency and are ready for further challenges.
• The Language Development sections for English Language Learners provides even more differentiated support for students who are at different levels of English proficiency.
• Community and Cultural Responsiveness activities build bridges between the mathematics students are learning to investigations of authentic contexts and issues. For example, students apply mathematics they have learned to objects and activities found in parks.

The materials also support learning with a variety of different experiences to develop connections in each lesson. For example:

• Lessons use manipulative objects such as counters and connecting cubes for hands-on experiences where students are actively applying and representing mathematics with hands-on objects.
• Partner and Whole group discussion are incorporated in all sessions to provide dialogue surrounding the mathematics being done.
• Visual representations and physical representations using mathematics manipulatives are used throughout each session.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for embedding tasks with multiple entry points that can be solved using a variety of solution strategies or representations. 

The materials present word problems at the beginning of most lessons in the Try It part of the session, which provides multiple points of entry for students. During Model It, students can model the problem in whichever way they like. Examples of problems that provide multiple entry points, different representations, and/or solution pathways include:

• In Grade 4, Lesson 27, Session 1, Explore, Try It states, “Kele and Kaci each buy equal-sized bottles of water. Kele now has 0.5 of his bottle left. Kaci has 0.4 of her bottle left. Who has more water left?” 

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations to engage students in regular and active participation in learning mathematics.

In Program Implementation, Implementation Support there are resources, including:

• A Multilingual Glossary
• A Bilingual Glossary
• An Academic Vocabulary Glossary
• WIDA PRIME V2 Correlation
• Try-Discuss-Connect Routine Resources

In Classroom Resources, Teachers Edition, Language Development is identified for each lesson. This differentiated instruction chart provides guidance for teachers at three levels of differentiation that identifies specific strategies (e.g., Speaking/Writing, Reading/Writing) directly connected to lesson sessions and activities. During sessions, specific strategies target additional supports for students to support engagement in lesson activities. For example:

• Prerequisite lessons are identified for most lessons and include specific supports for ELLs.
• Develop Language includes Why (rationale for the suggestions) and How (strategies and guidance on how to engage students) sections, along with explanations as needed.  
• Discuss It provides supports for all students to engage in mathematical discourse.
• Differentiated Instruction is included for most lessons and includes activities for intervention, on-level, and challenge.
• Math Center activities provide multiple levels of content. 
• Math in Action lessons in Grade 2 and higher build background in a variety of contexts to ensure access for all.

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing opportunities for proficient and advanced students to participate in enrichment activities for a deeper challenge. 

In i-Ready, Teach and Assess, Ready Classroom Mathematics, Classroom Resources, Teacher, Lesson Pacing Guide includes Small Group Differentiation Activities, and highlights Extend, Enrichment Activity. These Enrichment Activities can be found during the Refine session in the lesson, and include a Challenge Activity related to the content of the lesson. For example: 

• In Grade 4, In Lesson 31, Extend states, “Have students draw a circle using a compass. Have them draw a reflex angle with a measure of 200° by using a protractor to measure a 160° angle (360°- 160°= 200°) The larger angle formed measures 200°.”

The instructional materials reviewed for Ready Classroom Mathematics Grade 4 meet expectations for providing a balanced portrayal of demographic and personal characteristics representing a variety of backgrounds. 

In the Program Implementation tab, a multilingual, bilingual, and academic glossary are provided. Throughout the materials there are examples of animated pictures of children that have varied skin tone, features, hair color, and hair types such as pictures of children with brown skin and black hair, light skin and blond hair, and brown skin and brown hair. Pictures with more than one child show interactions between children with varied skin and hair colors such as one with light brown skin and black hair and the other with dark brown skin and black hair. The pictures of objects included are pencils, cars, soccer balls, footballs, apples, bananas, crayons, goldfish, rocks, flowers and other objects commonly known to most students. A variety of names representing different ethnic and cultural backgrounds are used, including:  Rosa and Ryan, Nick and Nora, Dave and Ari, Roberto and Rena, Fran and Pete, Gabe and Rose, Darious, Sam, Lexi, Paco, Julija, Lana, and Kyle.