4th Grade - Gateway 2

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.”