Into Math Florida

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Overall Summary

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for alignment to the Mathematics Florida Standards (MAFS). ​The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor. The materials partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Cluster Standards for Mathematical Practice (MPs).

Alignment
Meets Expectations
Usability
Meets Expectations

Focus & Coherence

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

Gateway 1
Meets Expectations

Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

​The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for not assessing topics before the grade level in which the topic should be introduced. The materials assess grade-level content and, if applicable, content from earlier grades.

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The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for assessing grade-level content. An Assessment Guide, included in the materials, contains two parallel versions of each Module assessment, and the assessments include a variety of question types. In addition, a Performance Task has been created for each Unit, along with Beginning, Middle, and End-of-Year Interim Growth assessments.

Examples of assessment items aligned to grade-level standards include:

• Unit 2 Performance Task states: “In Mexico, people use pesos for money. There are about 12.8 pesos in 1 dollar. About how much is 1 pesos worth in dollars? Show your work, and give your answer to the nearest hundredth of a dollar and the nearest cent.” (6.RP.1.3d)
• Module 4, Form A, question 13 states: “Jake has a wooden board that measures 8.25 feet long. He cuts the board into pieces that each measure 0.75 feet so he can paint signs. If Jake uses one piece per sign and sells each sign he makes for 15.95, how much money can he earn?” (6.NS.2.3) Above grade-level assessment items are present, but could be modified or omitted without a significant impact on the underlying structure of the instructional materials. These items include: • Unit 1 Performance Task, 1- 2: Students add positive and negative numbers which aligns to 7.NS.1.1. “The Number System Throughout History: 1) The ancient Chinese used counting rods….” Students are given a picture of sets of black and white rods to translate into numbers and solve, and the numerical expression is 5 + 3 + (-4) + 3 + (-5) + (-4); it could be solved with manipulatives. 2) “Negative numbers were used in the Islamic world to represent debts. The table below shows the purchases and payments that a customer makes at a merchant’s store during one month…” Students add 7 days worth of transactions, and the numerical expression is 8 + (-13) + (-5) + 15 + (-3) + (-8) + 10 + (-12). • Module 6, Form A, questions 2, 5, 7, 10, 12, 15: Six questions connect to Grade 7 Geometry (7.G.2.4) relating to circumference/radius/diameter of circles such as: “Jared ran around a circular track. The diameter of the track measured 98 meters. If he ran 4 laps around the track, approximately how far did Jared run? A) 306 meters B) 616 meters C) 1232 meters D) 2464 meters.” Note that Module 6, Form B has the same issue since the forms are parallel. Criterion 1.2: Coherence Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade. The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade. Indicator {{'1b' | indicatorName}} Instructional material spends the majority of class time on the major cluster of each grade. The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for spending a majority of instructional time on major work of the grade. • The number of Modules devoted to major work of the grade is 12 out of 16, which is approximately 75%. • The number of Lessons devoted to major work of the grade (including supporting work connected to the major work) is 46 out of 63, which is approximately 73%. • The number of Days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 105 out of 137 days, which is approximately 77%. A lesson-level analysis is most representative of the instructional materials because this calculation includes all lessons with connections to major work and isn’t dependent on pacing suggestions. As a result, approximately 73% of the instructional materials focus on major work of the grade. Criterion 1.3: Coherence Coherence: Each grade's instructional materials are coherent and consistent with the Standards. The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The instructional materials are also consistent with the progressions in the standards and foster coherence through connections at a single grade. Indicator {{'1c' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The instructional materials reviewed for Into Math Florida Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Examples of how the materials connect supporting standards to the major work of the grade include: • Lesson 2.3 connects 6.NS.2.4 with 6.NS.3.6 as students find and use the greatest common factor or least common multiple to compare rational numbers and solve various problems in multiple real-world situations. For example, “Cameron and Tatiana volunteer at the library. Cameron shelves one book every $$\frac{1}{4}$$ minute. Tatiana shelves one book every $$\frac{3}{10}$$ minute. Who is quicker at shelving books? Explain how you found your answer.” • In Lesson 5.2, students “Find the unit rate of5.15 per 5 pound bag” which connects 6.NS.2.3 with 6.RP.1.2.
• In Lesson 6.3, students multiply decimals (6.NS.2.3) to convert between measurement units (6.RP.1.3d) and write equivalent ratios. For example, “Many water bottles contain 16 fluid ounces, or 1 pint of water.  Drink labels often show the number of milliliters in a container. How many milliliters are in 16 fluid ounces?”
• In Lesson 8.5, students find an equivalent expression (6.EE.1.3) using the distributive property (6.NS.2.4).
• In Lesson 11.2, students graph ordered pairs on the coordinate plane (6.NS.3.6) to create polygons and solve related problems (6.G.1.3).
• In Lesson 11.4, students graph polygons on the coordinate plane (6.G.1.3) and solve multiple related problems including segment length, perimeter, and area (6.NS.3.8).
• In Lessons 12.1-12.4, students develop area formulas for various figures (6.G.1.1) and substitute numerical values into the formulas to evaluate the expressions (6.EE.1.2c).
• In Lesson 14.2, students summarize data in a dot plot (6.SP.1) and find the percentage of people that had more than 5 coins in their pocket or fewer than 7 coins in their pocket (6.RP.1.3c).

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The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials for Into Math Florida Grade 6 meet expectations that the amount of content designated for one grade-level is viable for one year. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications. As designed, the instructional materials can be completed in 161 days: 105 days for lessons and 56 days for assessments.

• The Planning and Pacing Guide and the Planning pages at the beginning of each module in the Teacher Edition provide the same pacing information.
• Grade 6 has five Units with 16 Modules, containing 63 lessons.
• The Pacing Guide designates 37 lessons as two-day lessons and 26 as one-day lessons, leading to a total of 100 lesson days; no information is provided about the length of a “day”.
• Each Unit includes a Unit Opener, which would take less than 1 day. There are five Openers for Grade 6 (five days).

Assessments included:

• The Planning and Pacing Guide indicates a Beginning, Middle, and End of Year Interim Growth assessment that would require one day each (three days).
• Each Module starts with a review assessment titled “Are You Ready?”. There are 16 Modules (16 days).
• Each Unit includes a Performance Task which indicates an expected time frame ranging from 25-45 minutes. There are five Performance Tasks for Grade 6 (five days).
• Each Module has both a review and an assessment. There are 16 Modules (32 days).
• Based on this, 56 assessment days could be added.

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Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials for Into Math Florida Grade 6 meet expectations for being consistent with the progressions in the Standards. In general, the materials identify content from prior and future grade-levels and relate grade-level concepts explicitly to prior knowledge from earlier grades. In addition, the instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems.

• In the Teacher Edition, the introduction for each Module includes Mathematical Progressions Across the Grades, which lists standards under the areas of Prior Learning, Current Development, and Future Connections and clarifies student learning statements in these categories. For example, in Module 5, Ratios and Rates: “Prior Learning: Students generate equivalent fractions.” (4.NF.1.1); “Current Development: Students understand the concept of ratio and use ratio language to describe a relationship between two quantities.” (6.RP.1.1); “Future Connections: Students will identify the unit rate given a table, verbal description, equation, or graph.” (7.RP.1.2)
• The beginning of each Module has Teaching for Success, which sometimes includes Make Connections. Make Connections references prior learning, for example in Module 9, “In the past, students have worked with numerical expressions and they should also have used words and phrases such as more than, divided by, and minus to translate numerical expressions into words. Here students use their prior knowledge of expressions to develop an understanding of basic equations and inequalities.”
• In Activate Prior Knowledge at the beginning of each lesson, content is explicitly related to prior knowledge to help students scaffold new concepts.
• Some lessons provide direct scaffolding for students reminding them of prior learning. For example, in Lesson 8.2, Step It Out states, “You learned about the order of operations in a previous grade. Now we need to add another step to the Order of Operations steps, exponents. Exponents are calculated after performing operations in parentheses or brackets.”
• In Unpacking the Standards, found in the Teacher Edition, there is discussion of how students use the skills they are learning in the future. For example, Lesson 1.1 states, “In the future, students will use this concept to fluently complete operations involving integers, including in algebraic representations of situations.”
• Each module includes a diagnostic assessment, Are You Ready?, it explicitly identifies prior knowledge needed for the current module. For example, in Module 2, Are You Ready? addresses finding equivalent fractions (4.NF.1.1), comparing fractions (4.NF.1.2), and comparing decimals (5.NBT.1.3b). In this module, students extend the skills to compare and order all rational numbers, including integers (6.NS.3.7).

Examples where standards from prior grades are not identified include:

• The Module Opener activities utilize standards from prior grade-levels, though these are not always explicitly identified in the materials. For example, in Module 3, students multiply fractions by whole numbers to find the number of pets Eliza owns (5.NF.2.4), and in Module 9, students write numerical expressions equaling a number on a dartboard (5.OA.1.1).

Examples of the materials providing all students extensive work with grade-level problems include:

• In the Planning and Pacing Guide, the Correlations chart outlines the mathematics in the materials. According to this chart, all grade-level standards are represented across the 16 modules.
• Within each lesson, Check Understanding, On My Own, and More Practice/Homework sections include grade-level practice for all students. Margin notes in the Teacher Edition also relate each On My Own practice problem to grade-level content. Examples include:
• In Lesson 1.3, Build Understanding Question 1D: ”Negative numbers are less than positive numbers. Does this mean that the absolute value of a negative number must be less than the absolute value of a positive number? Explain.” (6.NS.3.7d)
• In Lesson 11.2, Spiral Review Question 10: “A truck driver drives 245 miles and needs to drive m miles in all. Write an equation for the number of miles the driver has left to drive.” (6.EE.3.9)
• In Lesson 11.3, On My Own Problem Question 6: “What is the distance between point S and T? Are they in the same quadrant?” (6.NS.3.8, 6.NS.3.6b)
• When work is differentiated, the materials continue to develop grade-level concepts. An example of this is Lesson 4.1, which involves adding and subtracting multi-digit decimals. The corresponding Reteach page provides step-by-step notes and a process for students to follow in order to access the concept; the Challenge page provides students lists of times achieved by runners and swimmers in the Olympics and students must answer addition and subtraction questions involving each: “At the 2012 Olympics in London, the men’s 4 × 100 meter relay team from Jamaica set a world record. Individually, the times of the splits were 10.28 seconds, 9.07 seconds, 9.09 seconds, and 8.7 seconds. The official record was 36.84 seconds. A) What was the total of their individual times?  B) How much greater was the total of their individual times greater than the world record?”

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Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

The materials encompass learning objectives visibly shaped by CCSSM cluster headings, and examples of this include:

• In Lesson 2.3, the learning objective is “Compare rational numbers using the GCF and LCM”, and this is shaped by 6.NS.2.
• In Lesson 3.4, the learning objective is “Divide fractions and mixed numbers”, and this is shaped by 6.NS.1.
• In Lesson 13.2, the learning objective is “Find the volume of a rectangular prism”, and this is shaped by 6.G.1.
• In Lesson 9.4, two objectives are “Write and use equations to represent situations and solve problems” and “Describe the unknown quantity in a real-world situation; Explain why addition, subtraction, multiplication, or division should be used to model a situation”, and these are shaped by 6.EE.2.

The materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important, examples include:

• In Lesson 9.3, students solve equations using rational numbers (6.EE.2), requiring a fraction divided by a fraction (6.NS.1).
• In Lesson 10.1, students determine independent and dependent variables from a table of values (6.EE.3) and find the unit rate from the given data (6.RP.1).
• In Lesson 13.3, students find volume of rectangular prisms (6.G.1) given dimensions with decimal values (6.NS.2).

Rigor & Mathematical Practices

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for Gateway 2, rigor and balance and practice-content connections. The instructional materials meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and they partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

Gateway 2
Meets Expectations

Criterion 2.1: Rigor

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The instructional materials also do not always treat the aspects of rigor separately or together.

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Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

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)

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Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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)

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Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

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)

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Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

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?”

Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials reviewed for Into Math Florida Grade 6 partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified but not clearly labeled throughout the materials, and the instructional materials support the standards’ emphasis on mathematical reasoning.

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The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Into Math Florida Grade 6 partially meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade-level.

All MPs are identified throughout the materials, but they are not clearly labeled. There are inconsistencies in identifying the MPs in the materials, inaccurate identification, and over-identification of the MPs, and examples include:

• MPs are identified in both the Planning and Pacing Guide and the Teacher Edition, however they do not always align with each other. For example, in Lesson 9.2 about solving equations, the pacing guide identifies MP.1.1 while the Teacher Edition states MP.3.1, MP.4.1, and MP.7.1.
• The Planning and Pacing Guide explains each MP and provides a correlation to specific lessons, for example, the correlation for MP.2.1 can be found in, “In every ‘Spark Your Learning’ lesson and most lessons.”  MP.1.1 and MP.3.1 are correlated with “every lesson.”
• The Planning and Pacing Guide describes generally where to find the MPs, such as Spark Your Learning is always paired with MP.1.1, MP.3.1, and MP.5.1. This is different from previous identification which connects Spark Your Learning to MP.2.1. Connect Concepts and Skills focus on MP.7.1 and MP.8.1, sometimes MP.4.1; Apply and Practice addresses MP.2.1 and MP.6.1.

There are instances where MPs are naturally embedded and enrich the content, though not the majority of the time, and examples include:

• In the Teacher Edition lesson planning pages, MPs are identified in Lesson Focus and Coherence. The MPs are further identified within the lesson in Building Understanding and Step It Out. For example, Lesson 5.1 identifies MP.2.1, MP.3.1, and MP.6.1 as the lesson focus; then aligns Build Understanding with MP.2.1 and Step It Out with MP.6.1.
• Some lessons include an explanation about the connection to the MP in Professional Learning in the planning pages, for example, in Lesson 11.4, MP.8.1: “When calculating the perimeter of rectangles, students look for and express regularity in repeated reasoning by understanding that they need to add two sides and double the sum or double the lengths first, then add. Similarly, they look for and express regularity in repeated reasoning when they understand that they can multiply a square’s side length by 4 to find a square’s perimeter. In general, this standard includes finding shortcuts and devising new formulas when faced with regularly repeated calculations.”

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Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Into Math Florida Grade 6 partially meet expectations for carefully attending to the full meaning of each practice standard (MP).

The materials do not attend to the full meaning of MP.4.1 and MP.5.1. For MP.4.1, mathematical models are provided for students, and they use tools as directed by the materials, examples include:

• MP.4.1: In Lesson 6.2, Build Understanding, Task 2, students determine how many tons of concrete is 40,000 pounds of concrete, and are provided with scaffolded steps to complete the conversion. Pre-labeled ratios and equations are included in the scaffolded steps. Students substitute numbers into the pre-labeled ratios and complete computations based on the provided operations in the equations.
• MP.5.1: Throughout Modules 1 and 2, students use a number line to compare rational numbers. Number lines are provided, and students do not choose the tool as no other tools are presented as options. In Lesson 2.2, Question 2: “Diego is changing a recipe. Each ingredient is either reduced or increased. The changes to some of the ingredients are show below, in teaspoons: $$\frac{1}{8}$$ , -$$\frac{1}{4}$$ , 1 $$\frac{1}{2}$$ , -$$\frac{3}{4}$$. Complete the number line to compare the changes in the ingredients.” Students are provided with a number line from -1 to 2 with a mark at each $$\frac{1}{8}$$ teaspoon.

Examples of the instructional materials attending to the full meaning of the MPs include:

• MP.1.1: In Lesson 1.1, Spark Your Learning, “Fergal is recording the number of yards his school’s football team gained or lost on successive plays. How can you model or represent the opposite signs of each loss or gain shown in the table?” In Persevere, the Teacher Edition states, “If students need additional support, guide them by asking: What does it mean to gain yardage? What does it mean to lose yards? Are gaining and losing opposites?”
• MP.2.1: In Lesson 13.1, Question 3, students reason abstractly and quantitatively to answer, “Can the following nets be folded into cubes? If not, explain.”
• MP.6.1: In Lesson 3.3, Question 2, students attend to the precision of wording in the problem, “Jonathan will run a 6$$\frac{1}{4}$$-mile relay with 4 other team members, where each team member runs an equal distance. How many miles will Jonathan run?”
• MP.7.1: In Lesson 8.4, Question 10, students complete a table with the area of a triangle, given the height. Students use the structure of a triangle to determine, “How does the area change as the height increases? Why do you think this happens?”
• MP.8.1: In Lesson 8.1, students use regularity in repeated reasoning to: “Write 3^8 using repeated factors”, “Write $$3^3\times3^5$$ using repeated factors.”, and “Write the expression in exponential form: $$7\times7\times7\times7=$$."

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Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

An often-used strategy in these materials is Turn and Talk with a partner about the related task. Regularly, Turn and Talks require students to construct viable arguments and analyze the arguments of others. In addition, students are often asked to justify their reasoning in practice problems.

• In Lesson 2.4, Question 4 states, “D’Marc lists the sizes, in inches, of a set of screws: $$\frac{9}{64}$$, $$\frac{5}{32}$$, $$\frac{1}{16}$$, 18. He reasons that because the denominators are in order from greatest to least, the list is in order from least to greatest. Is D’Marc correct? Why or why not?”
• In Lesson 3.3, Question 14 states, “Dan says that 24 $$\frac{1}{2}$$ divided by 12 $$\frac{1}{2}$$ = 2, because $$\frac{24}{12}$$ = 2. Sam disagrees and thinks there will be fewer than 2 groups of 12 $$\frac{1}{2}$$ in 24 $$\frac{1}{2}$$. Who is is correct and why? What is 24 $$\frac{1}{2}$$ divided by 12 $$\frac{1}{2}$$?”
• In Lesson 4.4, Step It Out states, “Yan borrowed his parents’ car for a weekend camping trip. He drove the car 276.3 miles and used a total 10.230 gallons of gas. Kierra and Shawna both calculate how many miles Yan’s car drove per gallon. Whose solution is correct? Explain the error in the incorrect solution. How could you check the result?”
• In Lesson 5.1, Spark Your Learning, the Turn and Talk prompts, “Share your solutions. Did you use the same method? If not, explain your reasoning to make sure both methods are correct.”
• In Lesson 8.4, “Bill and Tia are trying to evaluate the expression $$5x^2$$ when x = 3. They both agree that 3 should be substituted for x. Tia says they should multiply 3 by 5, and then square the result. Bill says they should square 3 and then multiply by 5. Who is correct and why? What is the value of the expression?”
• In lesson 9.3, a Turn and Talk states, “Why can’t you divide both sides of an equation by zero? Explain.”
• In Lesson 12.1, Question 14 states, “For the two quadrilaterals below, Dan says that the one on the left has a larger area than the one on the right because it is longer. Bob says that both quadrilaterals have the same area. Who is correct? Why?”
• In Lesson 16.1, Question 3 states, “Construct Arguments. The dot plot shows the number of hours that 40 students studied each week. Make a statement that summarizes the data in the plot. Support your statement by describing clusters, gaps and/or peaks.”

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Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Many of the lesson tasks are designed for students to collaborate. Teacher prompts promote explaining their reasoning to each other during collaborative lesson tasks. Independent problems provided throughout the lessons also have teacher guidance to assist teachers in engaging students.

• The Teacher Edition provides Guided Student Discussion with questions to encourage students to explain their thinking. For example, in Lesson 4.1, “How is adding decimals similar to adding whole numbers? How is it different? How close to 10 kilometers is the length of your route?”, and in Lesson 14.3: “The movie theater recorded sales data for 14 days. How is this number represented in the histogram? Can you analyze the histogram to determine the total number of pretzels that the theater sold over the 14 days? Explain.”
• Turn and Talks are provided multiple times per lesson. For example, in Lesson 10.1, Task 3 Turn and Talk states, “Do the equations and the tables describe the same relationships? Explain how you know.” Teachers are given a possible answer as well as additional guidance to assist students in constructing arguments, for example, “If some students are having trouble understanding that the equations and tables show the same relationship, have students substitute the values for each row of each table into the related equation.”
• The Teacher Edition includes Let’s Talk in margin notes to prompt student engagement. For example, in Lesson 3.1, “Select students who used various strategies and have them share how they solved the problem with the class. Encourage students to ask questions of their classmates. Discuss how the division problem can be modeled with fraction strips, or how division can be calculated by multiplying by the reciprocal.”
• The Teacher Edition also provides Cultivate Conversation prompts in the lessons. For example, Lesson 2.4 includes, “Stronger and Clearer. Have students share how they solved the problem. Remind students to ask each other questions of each other that focus on how they approached the problem. Then have the students refine their answers.”
• In the margin notes for practice questions identified as a mathematical practice, an explanation about why that practice is labeled. For example, in Lesson 16.1, Question 3 is labeled Construct Arguments. In the Teacher Edition, the notes explain the problem “gives students an opportunity to demonstrate an understanding of clusters, gaps, and peaks to describe a data distribution in the context of a real-world situation.”
• In Lesson 6.2, Connect Math Ideas, Reasoning, and Language states, “Before beginning the task, have students describe and give examples on their own words where they might convert measurements, such as one foot to inches or 1 yard to feet. Have partners share their work and discuss how their descriptions compare and connect.”

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Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction on communicating mathematical thinking with words, diagrams, and symbols. The materials use precise, accurate terminology and definitions when describing mathematics and support students in using them. Examples are found throughout the materials.

• Key Academic Vocabulary is listed at the beginning of the module in a table including any prior vocabulary relevant to the lesson and new vocabulary.
• Each lesson includes a Language Objective emphasizing mathematical terminology. For example, in Lesson 10.1, “Use the terms dependent and independent to describe variables represented in equations, tables, and graphs.”
• In Module planning pages, a Linguistic Note on the Language Development page provides teachers with possible misconceptions relating to academic language. For example, in Module 1, “Listen for students who do not understand the meanings of the terms positive, negative, and opposite as they refer to numbers. Students may already know the words positive and negative in phrases such as positive attitude or negative thinker. Ensure that students understand that, in mathematics, positive and negative numbers don’t have a meaning of “good” or “bad.” Model the correct language for students.”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 3.5, students use a Frayer Model to define and explain the terms: least common multiple, greatest common factor, and common denominator. Students explain to each other how the terms are related.
• Guided Student Discussion often provides prompts related to understanding vocabulary such as, “Listen for students who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms origin, x-axis, y-axis, and ordered pair. Ask students to explain what they mean if they use those terms.”
• Student pages include vocabulary boxes defining content vocabulary.
• Vocabulary is highlighted and italicized within each lesson in the materials.
• The vocabulary review at the end of each Module require students match new vocabulary terms with their meaning and/or examples provided, fill-in-the-blank with definitions or examples, or create a graphic organizer to help make sense of terms.
• The Teacher Edition sometimes suggests creating an Anchor Chart to “connect math ideas, reasoning, and language” where students define terms with words and pictures, trying to make connections among concepts. For example, Lesson 13.1 shows a sample anchor chart including vocabulary related to nets, surface area, and volume.
• The Interactive Glossary at the end of the text provides the definition and a visual (diagrams, symbols, etc.) is provided for each vocabulary word. In the student book, the instructions read, “As you learn about each new term, add notes, drawings, or sentences in the space next to the definition. Doing so will help you remember what each term means.”

Usability

Gateway 3
Meets Expectations

Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for being well-designed and taking into account effective lesson structure and pacing. The instructional materials include an underlying design that distinguishes between problems and exercises, assignments that are not haphazard with exercises given in intentional sequences, variety in what students are asked to produce, and manipulatives that are faithful representations of the mathematical objects they represent.

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The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

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.

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Design of assignments is not haphazard: exercises are given in intentional sequences.

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.

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There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

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.”

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Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

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.

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The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials for Into Math Florida Grade 6 is not distracting or chaotic and supports students in engaging thoughtfully with the subject.

The entire series, both print and digital, follows a consistent format, which promotes familiarity with the materials and makes finding specific sections more efficient. The page layout in the materials is user-friendly. Tasks within a lesson are numbered to match the module and lesson numbers. Though there is a lot of information given, pages are not overcrowded or hard to read. Graphics promote understanding of the mathematics being learned. Student practice problem pages include enough space for students to write their answers and provide explanations. The digital format is easy to navigate, but students have to scroll without being able to view much of the information at one time.

Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for supporting teacher learning and understanding of the CCSSM. The instructional materials include: quality questions to support teachers in planning and providing effective learning experiences, a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials, a teacher edition that partially contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons, and explanations of the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

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Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

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?”

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Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

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.”

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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) 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.

The instructional materials for Into Math Florida Grade 6 partially meet the expectations for containing adult-level explanations so that teachers can improve their own knowledge of the subject. The materials include adult-level explanations of the grade-level content, but the materials do not include adult-level explanations of advanced mathematics concepts so teachers can improve their own knowledge of the subject. Examples of the grade-level explanations include:

• At the beginning of each module, the Teacher’s Edition includes Teaching for Depth providing a brief overview of the mathematics contained in the module. For example, in Module 5, “A ratio is a comparison of two quantities using division. In a classroom with 12 girls and 16 boys, the ratio of girls to boys is 12 to 16. The ratio may also be written as 12:16 or 12/16. As with fractions, equivalent ratios name the same comparison, and you can find equivalent ratios by multiplying or dividing both terms of a ratio by the same nonzero number. Thus, 12:16 is equivalent to 3:4. It is helpful for students to have different ways of describing and picturing ratios. For example, the ratio described above can be interpreted as follows: “For every 3 girls in the class, there are 4 boys.” This can also be represented visually, as in the figures below.” The figures show four red circle with three Gs in each one and four blue squares with 4 Bs in each one.
• In addition, Teacher to Teacher From the Classroom gives tips or anecdotes about the module content. For example, in Module 3, “When working on dividing fractions, I like to provide opportunities for students to make connections to what they already know. In fifth grade, students learn how to divide a fraction by a whole number and a whole number by a fraction. Knowing this, I engage my 6th grade students in discussions about fraction division that require them to build on that understanding. First, we discuss several 5th grade examples. I ask students to develop contextual situations for each problem. Next, we move into 6th grade examples, starting with a simple fraction division problem such as $$\frac{9}{15}\div\frac{1}{3}$$. If students developed a contextual situation for $$2\div\frac{1}{3}$$, for instance, we try out the same idea for $$\frac{9}{15}\div\frac{1}{3}$$.”

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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

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.

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Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials for Into Math Florida Grade 6 provide a list of lessons in the teacher's edition, cross-­referencing the standards addressed, and a pacing guide.

Each course in this series includes a Planning and Pacing Guide including the standards and pacing (number of days) for each lesson. Another standards chart is located in the Planning and Pacing Guide listing each standard and correlation to Student Edition Lessons. In the Teacher Edition, pacing is provided in the module planning pages, and the standards contained in each lesson are identified with written descriptions, as well as listed under Current Development in the Mathematical Progressions chart.

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Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials for Into Math Florida Grade 6 include strategies for parents to support their students progress. The Planning and Pacing Guide describes strategies to Connect with Families and Community:

• The student materials contain Math on the Spot problems with videos connected to them. “Math on the Spot video tutorials provide instruction of the math concepts covered and allow for family involvement in their child’s learning.” There are generally 1-3 problems per module.
• “Family letters inform families about the skills, strategies, and topics students are encountering at school.” Each module includes a letter, found online in 4 languages, providing vocabulary, a home activity, and discussion prompts.

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Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials for Into Math Florida Grade 6 explain instructional approaches used and how they are research-based.

The Planning and Pacing Guide contains Teacher Support Pages including a section on Supporting Best Practices. “Into Math was designed around research-based, effective teaching practices such as those described in Principles to Actions (NCTM 2014).” These include:

• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning and problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful mathematical discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.

The Planning and Pacing Guide describes four design principles from the Stanford Center for Assessment, Learning, and Equity (SCALE) to “promote the use and development of language as an integral part of instruction." These principles are: Support sense-making; Optimize output; Cultivate conversation; and Maximize linguistic and cognitive meta-awareness. To address this, the instructional materials include language routines to “help teachers embrace these principles during instruction.” Each module contains a Language Development page in the Teacher Edition stating where the language routines should be used. On the lesson pages of the Teacher Edition, Support-Sense Making boxes describe how the language routine can be used. Also, there are notes in the margin of the Teacher’s Edition provide connections from the strategy to the principle.

Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials reviewed for Into Math Florida Grade 6 partially meet expectations for offering teachers resources and tools to collect ongoing data about student progress on the CCSSM. The instructional materials provide strategies for gathering information about students’ prior knowledge, strategies for teachers to identify and address common student errors and misconceptions, and assessments that clearly denote which standards are being emphasized.

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Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The instructional materials for Into Math Florida Grade 6 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.

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Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The instructional materials for Into Math Florida Grade 6 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. A rationale explains the likely misunderstanding and suggests instructional adjustments or steps to help address the misconceptions.
• “Watch For” boxes and questions prompts that highlight areas of potential student misconceptions.

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Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials for Into Math Florida Grade 6 partially meet the expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

• Each lesson ends with 2-3 Spiral Review questions for ongoing practice in the More Practice/Homework section.
• Online interactive lessons and homework practice provide students with immediate notification about answers being correct or incorrect.
• The online lessons are the same as in the print textbook and provide immediate notification of correct or incorrect answers, but do not provide feedback for changing incorrect answers.
• Each Module Review has a scoring guide/checklist, so students know which questions they answer correctly. The scoring guide/checklist does not provide feedback for changing incorrect answers.
• Digital assessments are auto-scored and generate recommendations. They can provide feedback to teachers, but not directly to students.

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Materials offer ongoing formative and summative assessments:
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Assessments clearly denote which standards are being emphasized.

The instructional materials for Into Math Florida Grade 6 meet the expectations that assessments clearly denote which standards are being emphasized.

The standards alignment for each item on the Prerequisite Skills Inventory, Beginning-of-Year, Middle-of-Year, End-of-Year, and Module Tests are listed in the Assessment Guide on Individual Record Forms. Each Performance Task includes the standards in the teacher pages of the Assessment Guide, although the individual questions do not indicate which standards are being assessed.

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Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials for Into Math Florida Grade 6 partially meet the expectations that assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

• Each lesson has a diagnostic assessment, Are You Ready?, correlated to standards and a suggested intervention for struggling students. The materials state when using Online Ed, teachers can assign the Are You Ready? digitally “for immediate access to data and grouping recommendations.”
• “Check Understanding is a quick formative assessment in every lesson used to determine which students need additional support and which students can continue on to independent practice or challenges.” (Planning and Pacing Guide) Check Understanding presents a limited number of questions, usually 1-3, which includes a digital option that can be “auto-scored online for immediate access to data and recommendations for differentiation.”
• The Individual Record Forms in the Assessment Guide suggest Reteach Lessons teachers can use for follow-up based on the Module assessments, but there are no other suggestions for follow-up with students or guidance to teachers.
• The Individual Record Forms for the Prerequisite Skills Inventory, Beginning-of-Year, Middle-of-Year Test, and End-of-Year Tests do not suggest Reteach Lessons or provide other guidance teachers can use for follow-up with students.
• The Performance Task Rubrics for the Unit Performance Tasks do not suggest Reteach Lessons or provide other guidance teachers can use for follow-up with students.

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Materials encourage students to monitor their own progress.

The instructional materials for Into Math Florida Grade 6 include Scales to Track Learning Goals at the end of each lesson. The Teacher Edition introduction states, “The scales below can help you and your students understand their progress on a learning goal. Scales are also available in Module Resources.” The scale progresses from 1 to 4. For example from Grade 7, Lesson 1.1:

1. I cannot identify unit rate yet.
2. I can identify unit rates in tables but I still need help with writing the correct quantities in the numerator and denominator.
3. I can identify unit rates in tables by myself with few mistakes.
4. I can identify and use unit rates to complete tables and compare quantities without mistakes and explain it to others.

Each lesson includes “I’m in a Learning Mindset!” which gives students a prompt regarding the purpose of the lesson. For example, Perseverance: “What strategies do I use to stay on task when working on my own?”; Strategic Help-Seeking: “What is challenging about subtracting integers? Can I work through it on my own, or do I need help?”

Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The instructional materials reviewed for Into Math Florida Grade 6 meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners and strategies for meeting the needs of a range of learners. The materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations, and they provide opportunities for advanced students to investigate mathematics content at greater depth. The instructional materials also suggest support, accommodations, and modifications for English Language Learners and other special populations and provide a balanced portrayal of various demographic and personal characteristics.

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Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

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.

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Materials provide teachers with strategies for meeting the needs of a range of learners.

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.”

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Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

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.”

Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

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?”

Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

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.

Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

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.

Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for Into Math Florida Grade 6 provide opportunities for teachers to use a variety of grouping strategies.

• Each lesson provides teachers with a differentiated plan that includes small-group options.
• The materials provide students with self-directed activities at math centers.
• Throughout the materials, ample opportunities are provided for students to Turn and Talk with a partner.
• Using the Check for Understanding, the teacher is directed to pull students into small groups and use the Teacher Tabletop Flipchart.

Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials reviewed for Into Math Florida Grade 6 encourage teachers to draw upon home language and culture to facilitate learning.

• The student glossary is in both English and Spanish.
• Each Module includes School-Home Letters in multiple languages: Spanish, English, Portuguese, and Haitian Creole.

Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials reviewed for Into Math Florida Grade 6: integrate some technology in ways that engage students in the Mathematical Practices; are web-­based and compatible with multiple internet browsers; include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology; are intended to be easily customized for individual learners; and do not include technology that provides opportunities for teachers and/or students to collaborate with each other.

Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for Into Math Florida Grade 6 are web-based and compatible with multiple internet browsers.

• The materials are platform-neutral and compatible with Chrome, ChromeOS, Safari, and Mozilla Firefox.
• Materials are compatible with iPads, laptops, Chromebooks, and other devices connecting to the internet with an applicable browser. Online use was difficult on a Chromebook, there are scrolling and loading issues as well as difficulty seeing all pieces of the interactive editions.
• The materials are not compatible with an Android device (using Chrome browser). Although the website can be reached, it is not possible to zoom in or out, nor can one move the screen, so a student cannot access the entire screen.

Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for Into Math Florida Grade 6 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology through a website called Online ED, which parallels the print textbook. Only one Module per grade is currently available in the digital format, so some of the evidence is stated in the materials but has not actually been observed.

• Lesson problems from the Student Edition, assessments, and unit performance tasks are provided to be completed and scored using technology, providing students with feedback on whether the answers are correct or incorrect.
• Online Ed is designed to make recommendations for differentiation after auto-scoring of Check Understanding problems within each lesson.
• Growth monitoring assessments are “designed to be administered in 40 minutes, 3 times per year. The system utilizes a secure bank of assessments to adapt to each student’s ability and maps progress on the Quantile Framework.” (Pacing Guide)
• Assessments can be created using a question bank repeating the questions presented throughout the interactive lessons. However, teachers cannot modify questions nor add new questions.
• The online system has dynamic reporting by assignment or standards. If teachers are using the online system, they can view student progress for interim growth, module readiness, and lesson practice and homework.

Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for Into Math Florida Grade 6 are intended to include opportunities for teachers to personalize learning for all students. Full functionality of online materials is not accessible at the time of this review.

• Teachers can assign lesson problems and assessments, as well as view assessment analytics.
• Teachers can group students according to individual needs. The online component has Recommended Groups that “synthesizes data from assessments and places students into leveled groups.” (Pacing Guide) Recommended lesson resources can be assigned to each group.
• Teachers can create assessments using a bank of items.

The instructional materials reviewed for Into Math Florida Grade 6 provide minimal opportunity to be adapted for local use. Full functionality of online materials is not accessible at the time of this review.

• Pieces of a lesson can be assigned directly to students or groups of students.
• A question bank is provided for teachers to create assessments. The bank repeats the questions that are already included in each lesson, and these questions cannot be modified.

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials reviewed for Into Math Florida Grade 6 do not incorporate technology that provides opportunities for multiple students to collaborate with the teacher or one another.

Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for Into Math Florida Grade 6 integrate some technology, including digital lessons and virtual tools. Students can complete tasks and activities from the Student Edition through an interactive format.

• Students can draw pictures, create shapes, and type to show their thinking on the interactive lessons using a virtual sketchpad. Students complete tasks such as shading in the bar diagrams to represent $$\frac{5}{9}\div\frac{2}{9}$$, drag and drop the correct values into a table, or graph an equation. (Note: The backspace button, generally used to make a correction, is interpreted as the “back” button, returning to the previous screen and losing all work.)
• Only one Module per grade is currently available in the interactive lessons, so there is no way to know if the sketchpad is the only manipulative offered. No other virtual manipulatives were found.
• On the Spot videos of specific lesson problems are in the online student resources and provide the opportunity for students to review their work with their families by watching the video. These focus on content rather than MPs.

Report Overview

Summary of Alignment & Usability for Into Math Florida | Math

Math K-2

The instructional materials reviewed for Into Math Florida Grades K-2 meet expectations for alignment to the Mathematics Florida Standards (MAFS) and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

Kindergarten
Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations

Math 3-5

The instructional materials reviewed for Into Math Florida Grades 3-5 meet expectations for alignment to the Standards and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations

Math 6-8

The instructional materials reviewed for Into Math Florida Grades 6-8 meet expectations for alignment to the Standards and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations
Alignment
Meets Expectations
Usability
Meets Expectations

Overall Summary

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