The instructional materials for Into Math Florida Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions designed to develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Build Understanding and Step it Out introduce mathematical concepts, and students independently demonstrate their understanding of the concepts in Check Understanding and On My Own problems at the end of each lesson.

• In Lesson 5.1, students use ratio language to describe a relationship between two quantities such as birds visiting birdhouses, orange and black kittens, and colored fabrics in a quilt. For example, students “write a part-to-whole or whole-to-part comparison about the quilt using symbols and using ratio language, such as “for each”, “for every,” or “per”.” (6.RP.1.1, 6.RP.1.2)
• In Lesson 7.1, students use ratio tables to represent equivalent fractions leading to a percent of 100. Other visual representations include double number lines and place value blocks. In Lesson 7.2, students use a variety of strategies to find the percent of a whole, such as estimation, tape diagrams, and equivalent ratios. (6.RP.1)
• In Lesson 8.5, Step It Out, students substitute a value into two algebraic expressions to determine if the expressions are equivalent. This provides the understanding of different expressions can be equal. In the next task, students substitute missing values to rewrite an algebraic expression and identify the mathematical property used to rewrite the expression. Independent problems include similar exercises and provide students opportunities to build their own understanding. (6.EE.1.3)
• In Lesson 10.1, Build Understanding, students are presented with a situation represented in words, with a table, and with an equation. Students analyze the equation to show their understanding of the dependent and independent variables. In On My Own, students demonstrate understanding of the variables and the multiple representations showing the relationship between them by identifying the dependent and independent variable and completing a table and a graph representing the situation. (6.EE.3.9)

The instructional materials for Into Math Florida Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. The materials develop procedural skills and fluencies in On Your Own, and students demonstrate procedural skills and fluencies in More Practice/Homework.

• In Lesson 3.4, students practice division of fractions using the standard algorithm. For example, in Lesson 3.4, Question 9: “To paint her bedroom, Jade estimates she will need to buy 3 $$\frac{1}{4}$$ gallons of paint. How many $$\frac{1}{2}$$ gallon cans of paint should she buy? Explain.”; Question 5: “$$\frac{5}{4}$$ ÷ $$\frac{1}{10}$$”. (6.NS.1.1)
• In Module 4, students practice adding, subtracting, multiplying and dividing with multi-digit decimals using multiple strategies, including the standard algorithm. Expressions are presented both vertically and horizontally (6.NS.2.2 and 6.NS.2.3). Examples of problems for addition and subtraction include: 0.807-0.408; 0.13 + 0.58; “sum of 0.26 pound of red grapes and 0.34 pound of green grapes”; “difference between a cardinal weighing 1.5 ounces and a bluebird weighing 1.09 ounces.” Examples of problems for multiplication and division include: 10.05 × 5.6; 5104/116; 5.44 divided by 3.4; “the number of months it will take someone that pays $18 a month for a phone that costs a total of $432”; “how many shipping boxes are needed to ship 4,630 shirts when 130 shirts will fit in each box.”
• In Lesson 8.2, students evaluate exponents in numerical expressions and use order of operations, such as “$$72\div(15-6)+3\times2^2$$” and “$$15+32-6+(5+2)^2$$”, and in Lesson 8.4, students evaluate expressions by substituting values in for variables such as “Evaluate x − 12 when x = 18.6.” (6.EE.1.2c)

The instructional materials for Into Math Florida Grade 6 meet expectations for teachers and students spending sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and concepts of the grade-level, and students independently demonstrate the use of mathematics flexibly in a variety of contexts. During Independent Practice and On My Own, students often engage with problems including real-world contexts and present opportunities for application. More Practice and Homework contains additional application problems.

• In Lesson 5.3, students answer, “Two farms grow lettuce and tomatoes as shown in the labels to the right. Write ratios to compare acres of lettuce to acres of tomatoes at each farm. Which farm has a greater ratio of acres of lettuce to acres of tomatoes? How do you know?” In Lesson 5.5, Check for Understanding, Question 1 states, “A garden center is running a special on houseplants. A selection of 2 plants costs $7. If a designer buys 22 plants for new homes, how much does the designer spend on plants?” (6.RP.1.3)
• In Lesson 7.1, students solve real-world problems involving percent, such as, “Ryan got 36 out of 40 questions right on a test. Tessa got 92% on the same test. Who got a better score? Explain.” (6.RP.1.3)
• In Module 3, students solve word problems involving division of fractions by fractions such as, “Patrick has $$\frac{7}{10}$$ pound of flour. A batch of biscuits requires ⅛ pound of flour. How many whole batches of biscuits can Patrick make? Explain.” Also, “Daryl has $$\frac{2}{3}$$ of a bag of dog food. His dog eats $$\frac{4}{9}$$ of a bag per week. How many weeks will the dog food last?” (6.NS.1.1)
• In Module 9, students write and solve one-step equations for a variety of contexts, including differences in measurements and money. Examples respectively include, “Annie is 152.5 centimeters tall. She is 49 centimeters taller than her brother. Write and solve an equation to find her brother’s height in centimeters.” Also, “One ride on a city bus costs $1.50. Martina has $18 on her bus pass. Write and solve an equation to find how many rides she can take without loading more money on her bus pass.” (6.EE.2.7)

The instructional materials for Into Math Florida Grade 6 meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. In general, two or all three, of the aspects are interwoven throughout each module. The Module planning pages include a diagram showing the first few lessons addressing understanding and connecting concepts and skills, and the last lessons addressing applications and practice.

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

• Lesson 3.1 attends to conceptual understanding. Students use models to represent fraction division. For example, Part A) “Draw the fraction bar that you could use to begin to find the solution.” Part B) “Will the fraction bar you drew in Part A help you make groups of $$\frac{3}{8}$$? If not what other fraction bar could help? Explain why?” Then in part C, students use the new fraction bar to find out “How many groups of $$\frac{3}{8}$$ are in $$\frac{3}{4}$$? Explain.” The problems throughout the lesson ask students to use models to solve division of fractions or to write and solve problems based on the models presented.
• Lesson 15.2 develops procedural skill. After measures of center are defined, students calculate the mean, median, and mode in a variety of contexts including temperature, 40-yard dash times, and cat food consumption. For example, “The 40-yard dash time (in seconds) for 8 runners is shown. A) What is the mean of the data?  B) What is the median of the data?  C) What is the mode of the data?  D) Which of the measures of center has more than one possible value?”
• Lesson 2.4 emphasizes application of unit rates. Students find and interpret unit rates and apply the concept in a variety of contexts including: recipes, better buys, pool drainage, swimmers, painting, dog walking, reading a book, a model truck, entertainment, and more. An example problem includes, “Greg drove 300 miles at a constant speed in 4 hours. The speed limit was 70 miles per hour. Was he speeding?”

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

• Lesson 4.5 attends to procedural skills and application related to computing with multi-digit numbers and finding common factors and multiples. Problems in the lesson require application of the four operations with multi-digit decimals. For example, “Anwar waters his front lawn with a sprinkler that sprays 0.7 gallons of water per minute. If it takes 45 minutes to water the front lawn, how many gallons of water does Anwar use?” The lesson also requires students to demonstrate procedural fluency. For example, Questions 19-24 present students with decimal operation problems and they are instructed to “add, subtract, multiply, or divide.”
• Lesson 12.1 attends to all three aspects of rigor as students find the area of polygons. To build conceptual understanding, students derive the formula for the area of a parallelogram using visual representations and connecting to the area of a rectangle. The lesson includes multiple problems requiring students use the formula to find area of parallelograms using diagrams and given dimensions, such as “The height of a parallelogram is 4 times its base. The base measures 2 $$\frac{1}{2}$$ feet. Find the area of the parallelogram. Show your work.” Another example, “A window is in the shape of a rhombus, with each side being 20 in. long. The height of the window is 16 in. What is the area in square inches of the glass needed for the window?”

The instructional materials for Into Math Florida Grade 6 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 instructional materials for Into Math Florida Grade 6 meet the expectations that the design of assignments is intentional and not haphazard.

Overall, lessons are intentionally sequenced and scaffolded so students develop understanding 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 utilizing concrete and abstract representations and increase in complexity.

• Spark Your Learning serves to motivate and set the stage for students to learn new material and persevere through a related mathematical task.
• Build Understanding and Step It Out provide opportunities for students to learn and practice new mathematics, as well as “connect important processes and procedures” according to the Planning and Pacing Guide.
• Check Understanding provides a formative assessment opportunity after instruction.
• On My Own, More Practice/Homework, Test Prep, and Spiral Review in each lesson support students in developing independent mastery of the current lessons as well as reviewing material from previous lessons.
• Lessons are in a logical order and build coherence throughout the grade level.

The instructional materials for Into Math Florida Grade 6 meet the expectations for having a variety in what students are asked to produce, for example:

• Show written calculations and solutions
• Verbally defend or critique the work of others to show understanding
• Analyze double number lines and bar diagrams
• Build models for a problem by using diagrams and equations
• Use a diagram and a coordinate plane to represent a linear equation
• Compare multiple representations - table, graph, equation, situation - of data
• Use a digital platform to conduct and present their work
• Use manipulatives, especially in small groups, to represent mathematics
• Construct written responses to explain their thinking
• Performance Tasks: Grade 6, Unit 2, Trip to Mexico. “Denine took a trip to Mexico. While she was there, she had to use some of her math skills to understand distances and other units of measure and to exchange money.” Students use ratio reasoning to answer 4 questions about situations on the trip.
• STEM activities - Examples include: Grade 6, Unit 1, “Combinatorics is a branch of math focused on arrangements of objects. Jorge is an event organizer who must seat 102 guests. He has circular tables that seat 10 people each and square tables that seat 4 people each. He wants to use as many circular tables as possible but have no empty seats. Explain how he could arrange the tables he will need.”
  

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

• The series does not involve extensive use of manipulatives however, when they are included, they are consistently aligned to the expectations and concepts in the standards.
• Most hands-on manipulatives are integrated in supplemental, small-group, differentiated instruction activities and warm-up options.
• Examples of manipulatives include: Two-color counters, calculator, coins, number cubes, playing cards, string, square tiles, unit cubes, colored chips, algebra tiles, grid paper, index cards, anchor charts, ruler, compass, protractor, geometry software, bar diagrams, fraction strips, number lines, decimal grids, x-y tables, and pie charts.

The instructional materials for Into Math Florida Grade 6 meet the expectations for providing quality questions to help guide students’ mathematical development.

There are Guided Student Discussion questions and sample student answers throughout the Teacher Edition including on the Module opener page, Warm Up Options, Spark Your Learning, Build Understanding, Common Errors, and Step It Out pages corresponding to tasks or exercises on the page. Each module review also contains suggested questions intended to have students summarize concepts and skills developed within the module.

Each lesson introduction poses an essential question intended to guide student learning. For example, in Lesson 7.1, the Essential Question is, “How can you write a ratio as a percent?”

The Spark Your Learning planning page in the Teacher Edition includes examples of student work which show On Track, Almost There, and Common Errors. Each example has suggested questions for teachers to correct or advance student thinking. For example, in Lesson 2.3, Common Error about Least Common Multiples and Greatest Common Factors: “How does changing the denominator but not the numerator affect the value of a fraction?”

The instructional materials for Into Math Florida Grade 6 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, a variety of information is provided to 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 containing 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.

Two prompts in each module are related to Online Ed: “Assign the auto-scored Are You Ready for immediate access to data and grouping recommendations.” The other prompt being, “Assign the auto-scored Module Test for immediate access to data.” Within lessons, multiple prompts are presented: 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 6 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 discusses the “four progressions of 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 3-5, 6-7, and Grade 8/Algebra fit into these progressions.

The instructional materials for Into Math Florida Grade 6 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, notes, strategies, sample guided discussion questions, and possible misconceptions are provided 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 6 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

• Reteach and Challenge activities are located in each lesson.
• Each module includes Plan for Differentiated Instruction providing 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.

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 6 meet the expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

• Each Unit includes a STEM Task and a Unit Project which include multiple entry-points and a variety of solution strategies. Teachers are provided with possible answers, as well as What to Watch For tips, including: “Watch for students who become discouraged by a task and quickly give up. Strategies that may help these students include: working with a supportive partner, dividing the task into smaller steps, and reminding themselves that working at a difficult task is valuable, even if the task is not completed. Taking on new challenges is how we learn.” and “Watch for students who are reluctant to stretch themselves on a challenging task. Encourage these students to: identify similarities between the current task and tasks they have completed successfully in the past, identify one or more promising strategies or approaches, and try one of the strategies.”
• Each lesson begins with Spark Your Learning, an open-ended problem allowing students to choose their entry-point for applying mathematics and can be solved in a variety of ways. Suggestions in the Teacher’s Edition help students access the context of the problem. For example, in the side margin of the Teacher’s Edition, Motivate provides prompts such as in Grade 6, Lesson 9.1, “Introduce the problem. Point out that the problem does not state the amounts of money that Bella and Tia have, only that the two amounts are equal. Nevertheless, this information is enough to find a solution.”
• Support for Turn and Talk in the Teacher’s 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 6 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, further support in place for English Language Learners (ELLs) and other special populations:

• For ELLs, Language Development in each module includes linguistic notes providing strategies intended to help students struggling with key academic vocabulary such as: “Speak with students about words that can have multiple meanings….”, “Listen for students who do not distinguish between minus...and the negative sign.”, and “Visual cues help students…”
• Language Objectives are included in every lesson.
• Teacher Tabletop Flipchart Activities are referenced in the Teacher’s Edition for RtI support.
• Reteach, RtI Tier 2, and RtI Tier 3 worksheets can be assigned online or printed.
• Turn and Talk prompts are designed to support students in other special populations such as, “go back and reread the problem and break it into pieces. For example: What do you know? What do you need to find?”

The instructional materials for Into Math Florida Grade 6 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 and teachers may choose to utilize them 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 6 meet the expectations for providing a balanced portrayal of various demographic and personal characteristics.

• Lessons contain a variety of tasks that are of interest to students of various demographic and personal characteristics.
• Names and wording are chosen with diversity in mind. The materials include various names throughout the problems (e.g. Jayson, Suyin, Malik, Tressa, Anton, Jasmine, Yu, Felice, Sonia, Roselyn, Tracy, Tran, Arie, Miguel, Maria) and 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 not expressing gender, race, or ethnic bias, and there is no pattern in one character using more/fewer sophisticated strategies.
• When people are shown, a balance of demographic and personal characteristics.