4th Grade - Gateway 3

The instructional materials for Into Math Florida Grade 4 meet the expectations that there is a clear distinction between problems and exercises in the materials.

Each Module presents lessons with a consistent structure. During the instructional sections, which include Build Conceptual Understanding and Connect Concepts and Skills, students have opportunities to learn new content through examples and problems for guided instruction, step-by-step procedures, and problem solving.

At the end of the lesson, Apply and Practice provides a variety of exercises which allow students to independently show their understanding of the material. Exercises are designed for students to demonstrate understandings and skills in application and non-application settings. Test Prep and Spiral Review also include exercises.

The materials distinguish between problems and exercises within each lesson. Lessons include: Spark Your Learning, Build Understanding, Check Understanding, and On My Own. Spark Your Learning Problems activate prior knowledge and introduce new mathematics to students. Build Understanding includes problems that help students build conceptual understanding of the mathematics topic being taught. Step It Out sections help students to develop procedural skill and fluency.

Check Understanding and On My Own sections include exercises that ask students to use the newly learned mathematics in each lesson. Additional practice and Homework is available in a seperate student edition, providing more exercises for students to solve.

The instructional materials for Into Math Florida Grade 4 meet the expectations that the design of assignments is intentional and not haphazard.

Overall, lessons are intentionally sequenced and scaffolded so students develop in their understanding of mathematical concepts and skills. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skills, and engage with multiple activities that utilize concrete and abstract representations and increase in complexity.

Exercises are given in intentional sequences. In general, lessons are designed to begin with activating prior knowledge and build toward conceptual development and procedural skill. In the Spark Your Learning section of Lessons, students use manipulatives and/or visual models to experiment with the mathematics. Thus developing a concrete or representational understanding. This is followed by a Turn and Talk with a partner allowing students to process the connections they have found. Throughout the lessons, students are provided scaffolding with new content in the Build Understanding and Step It Out sections, where the abstract concept is broken down into smaller steps with additional turn and talk opportunities, and students are provided with independent exercises to build understanding and mastery. The Check Understanding section provides a mid-lesson check in and can be used to indicate the need to differentiate learning for students. Students practice the abstract concept in the On My Own.

The instructional materials for Into Math Florida Grade 4 meet the expectations for having a variety in what students are asked to produce.

In Spark Your Learning, Build Understanding, and Step It Out, students use visuals to show their thinking. Turn and Talk questions frequently ask students to construct arguments and give explanations. There are opportunities for students to produce answers and solutions in On My Own, while also providing opportunities for students to provide written explanations. Throughout the materials, students represent mathematics using equations.

Homework assignments ask for a variety of responses from fluency to higher level thinking. For example, the Lesson 9.1 Homework has six problems. The first problem asks students to construct an argument given constraints about the size of a piece of fabric if there is enough information to find the area. The next four problems are fluency problems with finding area and the last problem ask students to compare two sets of garden plans to determine which has the greater area.

The instructional materials reviewed for Into Math Florida Grade 4 meet the expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

At the beginning of the lesson, the materials indicate what materials/manipulatives will be needed, and the student pages include a picture of the manipulative they will use. For example in Lesson 4.1, students use two color counters to represent the following problem, “The fourth-graders make gift bags for the school bake sale. Each bag has 8 treats. How many treats do the students use to make 70 gift bags? Represent how many treats are in 7 bags.” The manipulatives provide opportunities for students to develop a conceptual model of problems that they will then represent in pictorial form in their student workbook.

Examples of manipulatives for Grade 4 include: Base Ten Blocks, connecting cubes, fraction circles, fraction strips, grid paper, number line, pattern blocks, protractor, ruler, scale, square dot paper, and two color counters.

Lesson 16.3, students create visual models with fraction strips or fraction circles to show multiplication of fractions by a whole number. They write equations to describe their visual models.

The materials rely on pictures of manipulatives. When physical manipulatives are used in the Lesson Materials in the Teacher Edition, it is not always clear how they are to be used. There is sometimes direction for how they can be used in Differentiation.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing quality questions to help guide students’ mathematical development.

Throughout the Teacher Edition questions are posted to help support teachers with questions to guide students’ mathematical development. Activate Prior Knowledge, Spark Your Learning, Build Understanding, Learn Together, and Turn & Talk, consistently provide questions to drive student discussion. For example:

• Lesson 8.6, Activate Prior Knowledge, “What is one strategy you could use when you multiply by 7?” In the same lesson, the Step It Out Turn and Talk states, “How can you use the Distributive Property to solve the problem?”
• Lesson 20.2, Spark Your Learning, includes questions in the margin notes, “What objects in the classroom are about 1 cm long? What objects are about 1 m long?” “How could you use classroom objects to act out the problem?” “What visual models can you draw to represent lengths or comparison?” and “What comparison words could you use to describe the length?”  
• Lesson 13.4, Build Understanding, provides two questions, “How could you figure out what angle measure is equivalent to one-fourth of a circle?” and “How could you figure out what angle measure is equivalent to two ninths of a circle?”

The instructional materials for Into Math Florida Grade 4 meet the expectations for containing ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.

In the Module planning pages, there is a variety of information that can help teachers understand the materials in order to present the content. Each lesson identifies the relevant content standards and Mathematical Practices, an Essential Question, Learning Objective, Language Objective, materials needed, and Mathematical Progressions Across Grades that contain prior learning, current development, and future connections. Unpacking the Standards provides further explanations of the standards’ connections. This section gives an explanation of the content standard contained in the lesson and Professional Learning, which sometimes contains information about the practice standard contained in that lesson. Teaching for Depth provides teachers with information regarding the content and how this relates to student learning.There are additional suggestions about activating prior knowledge or identifying skills in Warm-up Options, activities to Sharpen Skills, Small-Group Options, and Math Centers for differentiation.

There are two prompts in each module related to Online Ed: “Assign the auto-scored Are You Ready for immediate access to data and grouping recommendations.” and “Assign the auto-scored Module Test for immediate access to data.” Within lessons, there are multiple prompts: Warm-Up Options and Step It Out both have an icon, “Printable & projectible.”; “More print and digital resources for differentiation are available in the Math Activities Center.”; and “Assign the auto-scored Check Understanding for immediate access to the data and recommendations for differentiation.”

The instructional materials for Into Math Florida Grade 4 meet the expectations for explaining the role of the grade-level mathematics in the context of the overall mathematics curriculum.

Each module in the Teacher Edition includes Mathematical Progressions Across the Grades which lists prior learning, current development, and future connections. Similarly, the beginning of each lesson in the Teacher Edition includes Mathematical Progressions showing connections to prior and future grades’ standards, as well as other lessons within the program.

In the Planning and Pacing Guide, Progressions and Algebra Readiness notes “Algebra as a course of study today is integrated around four progressions of elementary and middle school content leading to the Algebra course: Number and Operations, Operations and Algebraic Thinking, Statistics and Probability, and Functions” and includes a table showing how the domains in Grades K-5, 6-7, and Grade 8/Algebra fit into these progressions.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing strategies for gathering information about students’ prior knowledge within and across grade levels.

• At the beginning of the year, students’ prior knowledge is gathered through a Prerequisite Skills Inventory. “This short-answer test assesses core precursor skills that are most associated with on-grade success.” (Assessment Guide)
• Each Module begins with Are You Ready, a diagnostic assessment of prior learning related to the current grade-level standards. Intervention materials are provided to assist students not able to demonstrate the necessary skills. Commentary for each standard explains how the prior learning is relevant to the current Module’s content.
• Prior learning is identified in the Mathematical Progressions section at the beginning of each Module and lesson of the Teacher Edition.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

• The module overview in the Teacher Edition contains “Common Errors” as students engage in an introductory task and provides questioning strategies intended to build student understanding.
• The Spark Your Learning planning page for each lesson in the Teacher Edition includes a Common Error section related to the content of the lesson identifying where students may make a mistake or exhibit misunderstanding. There is a rationale that explains the likely misunderstanding and suggests instructional adjustments or steps to help address the misconceptions.
• There are also “Watch For” boxes and question prompts highlighting areas of potential student misconceptions.

The instructional materials for Into Math Florida Grade 4 meet the expectations that assessments clearly denote which standards are being emphasized.

The Lesson Focus and Coherence page indicates the CCSSM that will be addressed within the lesson. Throughout the lesson are formative assessments in the Check for Understanding, On My Own, and More Practice/Homework. Each Module has an End of Module Test, the standards associated with each problem on this test can be found on the Individual Record Form within the Assessment Guide Book.

Each Unit has a summative Performance Task including the standards in the teacher pages of the Assessment Guide, although the individual questions do not indicate which standards are being assessed.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• At the beginning of each module, Teaching for Depth provides information on strategies to use when teaching the concept, including Represent and Explain, which focuses on ways for students to describe and picture a concept, or Make Connections, which helps students understand a new idea by connecting it to previous knowledge.
• At the beginning of each module, Mathematical Progression Across the Grades makes connections to both prior and future skills and standards to scaffold instruction.
• At the beginning of each module, Diagnostic Assessment, Are You Ready?, allows teachers to “diagnose prerequisite mastery, identify intervention needs, and modify or set up leveled groups.”
• Each lesson provides Warm-up Options to activate prior knowledge such as Problem of the Day, Quick Check for Homework, and Make Connections.
• Throughout the lessons, there are notes, strategies, sample guided discussion questions, and possible misconceptions that provide teachers structure in making content accessible to all learners.
• Student practice starts with up to four Check Understanding exercises to complete with guidance before moving to independent work in On My Own or More Practice/Homework.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

• There are Reteach and Challenge activities for each lesson.
• Each Module includes Plan for Differentiated Instruction that provides teachers with teacher-guided, Small-Group Options and self-directed Math Center Options based on student need, “On Track/Mixed Ability, Almost There (RtI), and Ready for More.”
• Each lesson provides Leveled Questions in the Teacher’s Edition identified as DOK 1, 2, and 3 with an explanation of the knowledge those questions uncover about student understanding.

There are three “Language Routines to Develop Understanding” used throughout the materials: 1) “Three Reads: Students read a problem three times with a specific focus each time.” 2) “Stronger and Clearer Each Time: Students write their reasoning to a problem, share, explain their reasoning, listen to and respond to feedback, and then write again to refine their reasoning.” and 3) “Compare and Connect: Students listen to a partner’s solution strategy and then identify, compare, and contrast this mathematical strategy.”

The instructional materials for Into Math Florida Grade 4 meet the expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The Planning and Pacing Guide, Teacher Support, Access and Equity, and Spark Your Learning Tasks are “designed as ‘low-floor/high ceiling’ tasks that all students can access but that can also be extended to provide challenge.”  Teachers are provided guidance on how to assist various levels of learners, depending on how they respond to the problem. For example, Lesson 5.4, Spark Your Learning has this prompt, “The Monstrosity Roller Coaster has seats for 136 riders.  The roller coaster completes 4 runs each half hour. If all the seats on the roller coaster are filled, how many people can ride in a half hour.” This problem provides multiple entry points and solution strategies for students. However, Spark Your Learning is not present in every lesson.

Support for Turn and Talk in the Teacher Edition provides suggestions to help students using a variety of strategies. Teachers are often prompted to “Select students who used various strategies and have them share how they solved the problem with the class.”

The instructional materials for Into Math Florida Grade 4 meet the expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

In addition to the strategies for meeting the needs of a range of learners described in Indicator 3s, there is further support in place for English Language Learners (ELLs) and other special populations:

There is Language Development to support English Learners in each module which includes linguistic notes that provide strategies intended to help students struggling with key academic vocabulary such as: “Speak with students about words that can have multiple meanings…," and “Visual cues help students…” Language Development also includes information about the Language Routines embedded in the instructional materials: Three Reads; Stronger and Clearer Each Time; Compare and Contrast; Critique, Correct, and Clarify. These are identified by a pink box throughout lessons with speech bubble that identifies the Language Routine to be used.  In addition, there are supports for special populations including:

• Language Objectives are included in every lesson.
• Reteach and RtI worksheets that can be assigned online or printed.
• Turn and Talk prompts designed to support students, for example, “go back and reread the problem and break it into pieces. For example: What do you know? What do you need to find?”
• A multi-lingual glossary is available online.

The instructional materials for Into Math Florida Grade 4 meet the expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

In addition to the strategies for meeting the needs of a range of learners described in Indicator 3s, there is further support in place for advanced students:

• Optional lessons are provided online. Teachers may choose to utilize with advanced students.
• Each lesson has a corresponding Challenge page, provided in print or online, addressing the same concepts and standards where students further extend their understanding and often use more complex values in their calculations.
• On the Module opener page, Extend the Task in the margin of the Teacher’s Edition provides ideas for extending the task.

The instructional materials for Into Math Florida Grade 3 meet the expectations for providing a balanced portrayal of various demographic and personal characteristics.

• Lessons contain a variety of tasks that interest students of various demographic and personal characteristics.
• Names and wording are chosen with diversity in mind. The materials include various names throughout the problems that are used in ways that do not stereotype characters by gender, race, or ethnicity.
• When multiple characters are involved in a scenario, they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias, and there is no pattern in one character using more/fewer sophisticated strategies.
• When people are shown, there is a balance of demographic and personal characteristics.