The instructional materials for HMH Into Math Grade 5 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, there is a Performance Task for each unit, and there are Beginning, Middle, and End-of-Year assessments.

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

• Unit 1, Performance Task, Questions 2 and 3, students find volumes of rectangular prisms. “The factory makes boxes that each hold one candle. The boxes measure 1 inch x 1 inch x 9 inches. The factory owner makes a stack of 40 candle boxes. What is the volume of the stack? Show your work.” (5.MD.5a)
• Module 2, Form A, Question 10, “A school earned $2,604 selling tickets to a fundraising event. If each ticket cost $14, how many tickets were sold?” (5.NBT.6)
• Module 6, Form A, Question 8, students solve a story problem by adding fractions with unlike denominators. (5.NF.2)
• Module 7, Form A, Questions 6-8, students add or subtract fractions and mixed numbers. (5.NF.1)
• Module 10, Form A, Question 4, students shade a model to represent the quotient of $$\frac{1}{6}\div2$$. (5.NF.7a) 
• Module 12, Form A, Question 3, “Ms. Yang left work at 5:15 p.m. She went to the gym for 90 minutes, and then it took her 40 minutes to drive home. What time did Ms. Yang get home?” (5.MD.1)
• End-of-Year-Test, Question 15, students find the difference in length between the longest and the shortest lions using a number line with fractional units. (5.MD.2)

The instructional materials reviewed for HMH Into Math Grade 5 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 15 out of 20, which is approximately 75%.
• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 83 out of 96, which is approximately 86%.
• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 135 out of 176 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 is not dependent on pacing suggestions. As a result, approximately 86% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for HMH Into Math Grade 5 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 12.4, 5.MD.1 supports the major work of 5.NF.4, multiplication of fractions. Students multiply fractions and mixed numbers by whole numbers to convert among units of measure. 
• Lesson 12.3, 5.MD.2 supports the major work of 5.NF.1, add and subtract fractions with unlike denominators. Students make line plots with fractional units, then use the line plot to answer questions such as “What is the total number of cups of water?”
• Lesson 18.1, Question 9, 5.MD.1 supports the major work of 5.NBT.2, multiplying and dividing with powers of 10. Students shift the decimal point to either multiply or divide by a power of 10 to solve conversion problems. “Convert 58 g to kg, and question 10 Convert 6257 cL to L.”

The instructional materials for HMH Into Math Grade 5 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 166 days, 115 days for lessons and 51 days for assessments.

• The Planning and Pacing Guide and Planning pages at the beginning of each module in the Teacher Edition provide the same pacing information. 
• Grade 5 has 8 units with 20 modules that contain 96 lessons. 
• The Planning and Pacing Guide designates 11 lessons as 2-day lessons and 85 as 1-day lessons, leading to a total of 107 days. The materials do not define the number of minutes in a lesson or instructional day. 
• Each unit includes a Unit Opener, and there are 8 unit openers for Grade 5 (8 days).
• Each lesson includes a variety of supplemental instruction such as reteaching lessons, flipbook lessons, etc. However, there is no guidance around building in days for differentiation; therefore, no additional days were added.
• This is a total of 115 lesson days.

Assessments included: 

• The Planning and Pacing Guide indicates a Beginning, Middle, and End-of-Year Interim Growth assessment that would require 1 day each (3 days). 
• Each unit includes a Performance Task which indicates an expected time frame ranging from 25-45 minutes. There are 8 Performance Tasks for Grade 5 (8 days). 
• Each module has both a review and an assessment. There are 20 modules (40 days). 
• Based on this, 51 assessment days can be added.

The instructional materials for HMH Into Math Grade 5 meet expectations for the materials being consistent with the progressions in the Standards. In general, the materials identify content from prior and future grade-levels as well as relating 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. 

The introduction for every module in the Teacher Edition includes Mathematical Progressions which lists standards under the areas of Prior Learning, Current Development, and Future Connections, as well as clarifying student learning statements in these categories. For example, at the start of Module 4, Prior Learning is listed as “Interpreted a multiplication equation as a comparison,” “represented verbal statements of multiplicative comparison as a multiplication equations,” and, “solve word problems involving multiplicative comparisons.” (4.OA.1 and 4.OA.2) Future Connections are listed as “will write numerical expressions involving whole-number exponents,” “will identify parts of an expression using mathematical terms,” and “will evaluate algebraic expressions for specific values of their variables using the order of operations.” (6.EE.1, 6.EE.2b, 6.EE.2c) Additional features of the materials further support the progressions of the Standards. These include:

• The beginning of each module includes a diagnostic assessment called Are You Ready? that explicitly identifies prior knowledge needed for the current module. 
• Within each lesson, the standard of focus is explicitly connected to work in future and prior grades. For example, Module 14, Lesson 3 identifies 5.NBT.7 as the focus for the lesson. The mathematical progressions, indicated in the Teacher Edition, show this builds upon work done with 4.NBT.4 during Grade 4 in Module 2, Lessons 1-3. This work will continue in Grade 6 with a focus on standard 6.NS.3. during Module 4, Lesson 1.

The materials give all students extensive work with grade-level problems. Students spend 4-8 days within each module and one day per lesson. Each lesson includes a Problem of the Day to activate prior knowledge, a Spark Your Learning portion as an introduction to the day’s learning goals that usually embeds partner or group work to solve a problem. Each lesson includes grade-level work in the Build Your Understanding, Step It Out, and On Your Own Sections.  Additionally, Reteach and Challenge pages are available for each lesson which provide more practice with grade level work. 

For example: 

• Lesson 15.1, Build Understanding, students find place value patterns when multiplying by powers of 10. Question 1C “In which direction do the digits shift as you multiply by increasing powers of 10?” The On Your Own section has 23 questions that have students practice multiplication of whole numbers and decimals with powers of 10.  For example: Question 12: 843 = 100 x ___. (5.NBT.2)  
• Lesson 2.1, Build Understanding, Question 1, students use rectangular arrays, multiplication equations and related division equations to solve “A local theater group is performing a musical. The members arrange 105 chairs in 5 equal rows for the audience. How many chairs are in each row?” (5.NBT.6)
• Lesson 7.2, On Your Own, includes 17 grade level problems for students to practice addition and subtraction of fractions with unlike denominators. For example, Question 10 states “Mr. Braxton’s laptop memory is $$\frac{9}{10}$$ full. After deleting unneeded files, the memory is $$\frac{2}{3}$$ full. Mr. Singh’s laptop memory is $$\frac{9}{10}$$ full. After he deletes some files, the memory is $$\frac{3}{5}$$ full.  What fraction represents the part of the laptop memory that he deleted? Is your answer reasonable? How do you know?” (5.NF.1) 
• Lesson 13.4, students expand upon their learning of place value and comparing and ordering numbers  to decimals. (4.NBT.1 and 4.NBT.2) Question 4, students evaluate four percentages written as decimals and order them from least to greatest. (5.NBT.3b) 

The materials relate grade-level concepts to prior knowledge from earlier grades. The Teacher Edition clearly identifies the previous grade level work and explains how students will use these skills in upcoming lessons. For example:

• In the Activate Prior Knowledge section at the beginning of each lesson, content is explicitly related to prior knowledge to help students scaffold new concepts.
• Module 7, Are You Ready shows the link to prior learning for Explore Mixed Numbers as Grade 4, Lesson 15.2 (4.NF.3b) in the Data-Driven Intervention Chart. A narrative is provided for each skill on the page. In Explore Mixed Numbers, the items assess whether students are able to find the mixed number equivalent to a given fraction greater than 1. “In the upcoming lessons, students will use this skill when adding and subtracting mixed numbers.”
• Module 14, Lesson 14.3, Assess Reasonableness of Sums and Differences, asks students to build upon their prior learning of understanding “fluently added multi-digit numbers (Grade 4, Lessons 2.1 and 2.3)” and “fluently subtracted multi-digit whole numbers (Grade 4, Lessons 2.2 and 2.3.).” The lesson additionally includes a Make Connections section where it suggests Project the Interactive Reteach, Grade 4, Lesson 2.1. and Complete the Prerequisite Skills Activity where a problem is presented to the students. They are asked to identify the operation used to solve the problem and explain how they decided which operation to use.

The instructional materials reviewed for HMH Into Math Grade 5 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

The materials include learning objectives that are visibly shaped by CCSSM cluster headings. Examples include:

• The Module 19 learning objective Graphs and Patterns is shaped by the cluster heading 5.OA.2: Analyze Patterns and Relationships and 5.G.1: “Graph points on the coordinate plane to solve real-world and mathematical problems.” 
• In Lesson 8.4 the learning objective is “Use a visual model to represent multiplication of fractions” which is shaped by 5.NF.2: “Apply and extend previous understandings of multiplication and division to multiply and divide fractions.”
• In Lesson  3.1 the learning objective is “Divide whole number dividends by 2-digit-divisors to find quotients with remainders” which is shaped by 5.NBT.2: “Perform operations with multi-digit whole numbers and with decimals to hundredths.”
• In Lesson 7.5 the learning objective is “Add fractions and mixed numbers with unlike denominators using properties” which is shaped by 5.NF.1: “Use equivalent fractions as a strategy to add and subtract fractions.” 

The materials include problems and activities that 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, and examples of this include:

• Lesson 5.6 connects 5.MD.3 with 5.NBT.2 when students use multi-digit multiplication to find the volume of composed figures. Question 11, students find the volume of two connected rectangular prisms with dimensions $$12 \times 50 \times 30$$ and $$10 \times 5 \times 15$$.
• Lesson 9.4 connects 5.NF.4b with 5.NBT.6 when students use fractional side lengths to solve real world problems with fraction multiplication. On My Own, Question 9 states “A square window has side lengths that are . . .  What is the area of the window?”
• Lesson 8.3, Represent Multiplication with Unit Fractions, connects 5.NF.2 with 5.NBT.2. For example, Question 2b, students use what they know about multiplication equations to model a problem. 
• Module 19 connects graphing in the coordinate plane (5.G.1) with generating two numerical patterns. (5.OA.2)

The instructional materials for HMH Into Math Grade 5 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Throughout the materials, there are sections that emphasize introducing concepts and developing understanding such as Build Understanding and Spark Your Learning. Students have the opportunity to independently demonstrate their understanding in the Check Understanding and On Your Own problems at the end of each lesson. Evidence includes: 

• Lesson 2.3, Spark Your Learning states “One of Miami’s tallest buildings is 900 Biscayne Bay. It stands 650 feet tall. The building has 63 floors. If each floor is approximately the same height, about how tall is one floor of the 900 Biscayne Bay building?” (5.NBT.6)
• Lesson 3.4, On My Own, states “Tina’s taco truck sold 3 times as many veggie tacos in August than in September. She sold twice as many in September than in October. She sold 927 veggie tacos during the three months. Represent the number of veggie tacos sold with a bar model. Write an equation to show the amount represented by each box of the bar model. Then find the amount. How veggie tacos did Tina sell in September? Explain how you know.” (5.NBT.6)
• Lesson 8.1, Check for Understanding, Question 1 states “At nine o’clock, $$\frac{5}{8}$$ of the 16 cats at the party go home. How many cats go home at nine o’clock? Draw a visual model to find the answer.” (5.NF.4)
• Lesson 8.5, Spark Your Learning states “A contractor buys rectangular floor tiles for a home that he is building. How can you find the area of the tile? Use the square to find the area of the tile. Explain your reasoning.” (5.NF.4) 
• Lessons 10.5, On My Own, Problem 11 states “Mae uses the expression $$5\div\frac{1}{6}$$ to solve a problem. Write a word problem that can be modeled by the expression. Draw a visual representation to show the quotient. Write an equation to model the problem. Interpret the quotient for the situation.” (5.NF.7)
• Lesson 13.2, On My Own provides students the opportunity to reason on what decimal has “$$\frac{1}{10}$$ of the value of 0.08 and 10 times as much as the value of .008? Explain.” (5.NBT.1)
• Lesson 14.4, More Practice and Homework, Question 6, Math on the Spot states “Tania measures the growth of her plant each week. The first week, the plant’s height measured 2.65 decimeters. During the second week, Tania’s plant grew .7 decimeter. How tall was Tania’s plant at the end of the second week? Describe the steps you took to solve the problem.” (5.NBT.7)
• In Lesson 15.2, students use visual models and base ten representations to represent multiplication with decimals and whole numbers. (5.NBT.7)
• In Lesson 15.5, students use an area model to multiply decimals by decimals. (5.NBT.7)
• Lesson 6.1, students utilize visual models such as fraction strips/bar model to add fractions with different denominators. (5.NBT.7)

The instructional materials for HMH Into Math Grade 5 meet expectations that they attend 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. Procedural skills and fluencies are intentionally built on conceptual understanding. This is primarily found in two areas of the materials: 

• In the On Your Own section, students work through activities to practice procedural skill and fluency.
• In the More Practice/Homework section, students can find additional fluency practice.  

Students have numerous opportunities to develop and independently demonstrate procedural skill and fluency, especially where called for by Standard 5.NBT.5. Examples include but are not limited to: 

• In Lesson 2.4, students use partial quotients to solve multi-digit division problems. On Your Own, Problem 8, “2,352 ÷ 48”. In the More Practice and Homework, Problem 6 “8,632 ÷ 29.” (5.NBT.6)
• The Module 3 Review provides students practice with division. In Problems 8 and 9, students determine whether an estimated digit is too high or low, adjusting the estimate if needed. Then they divide. Problem 8. “The estimate is 90. 8,645 ÷ 91.” Problem 9, “The estimate is 60. 1,243 ÷ 21.”
• In Lesson 8.3, On Your Own, students write an equation before multiplying. Problem 1, “$$1\frac{1}{4}$$ by $$1\frac{1}{3}$$.” (5.NBT.5)
• Lesson 8.5 presents opportunities for students to practice fluency with multiplication of fractions in both the Check for Understanding and On Your Own section. For example, Problem 4, “Find the product. $$\frac{4}{9}\times\frac{3}{5}$$”. Problem 9 “$$\frac{3}{8}\times\frac{3}{7}$$.” (5.NF.4)
• In Lesson 14.5, On Your Own, students practice subtracting decimals. For example, in Problem 15, students are asked “Find the Difference, 27.64 - 16.98” and in Problem 18, students are asked “Find the unknown number, 1.7 = __ - 4.63.” (5.NBT.7)

The instructional materials for HMH Into Math Grade 5 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level. Students also have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Application contexts are used throughout the curriculum to build conceptual understanding. During Independent Practice and On Your Own, students often engage with problems that include real-world context and present opportunities for application. The More Practice/Homework sections also contain additional application problems. 

• Lesson 3.2, On Your Own, Problem 13 states “Mr. Torres has 212 coins in his collection. He wants to keep all of his coins in a binder. He can store 24 coins on each binder page. He finds 212 ÷ 24 to be 8, r20, so he buys 8 pages. Does Anderson buy the correct number of pages? Explain.” (5.NBT.6)
• Lesson 8.4,On Your Own, Problem 3, Model with Mathematics: Write a story problem that can be modeled with the equation $$\frac{1}{4} \times \frac{8}{12} = \frac{2}{12}$$. Then draw a visual model to represent the problem.” (5.NF.6)
• Lesson 8.7, On your Own, Problem 10, students solve real world problems involving multiplication of fractions and mixed numbers. “Sam is using craft felt to carpet two rooms in a dollhouse. Both rooms are $$\frac{5}{6}$$ ft by $$\frac{7}{8}$$ foot. How many square feet of craft felt does Sam need to carpet both rooms? Explain your reasoning.” (5.NF.6) 
• Lesson 9.2, More Practice/Homework, Problem 1, students solve real world problems involving multiplication of fractions and mixed numbers. “Trail running is an exercise that involves running on trails instead of paved roads to reduce the impact on ankles and knees. Samantha runs on the Lakeside Trail. She runs $$2\frac{1}{2}$$ times around the loop and then walks the remainder of the way. Write an equation to model the distance Samantha runs.” (5.NF.6)
• Lesson 9.1, On Your Own, Problem 5, students are given the dimensions of a square tile used to tile a rectangular-shaped patio and to ask students to choose a length and width for the patio given constraints. Then students find the area of the patio and explain the method used. (5.NF.6 )
• Lesson 9.4, On Your Own, Problem 4, students solve real world problems involving multiplication of fractions and mixed numbers. “The area of a bathroom is 40 square feet. The area of another room is $$2\frac{3}{4}$$ times as great as the area of the bathroom. What is the area of the other bedroom?”  (5.NF.6)
• Lesson 9.1, Problem of the Day, students solve real world problems involving multiplication of fractions and mixed numbers as they determine how much water would go into 4 beakers if they each held $$2\frac{2}{8}$$ liter of water. Students are prompted to draw a model. (5.NF.6)
• Lesson 10.3, On Your Own, Problem 7, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “Consider the expression $$\frac{1}{5} ÷ 3$$. Write two different word problems that can be represented by this expression. Draw a visual model to represent the problems and then solve. What does the quotient represent in each problem?”  (5.NF.7c) 
• Lesson 11.2, On Your Own, Problem 13, “Maggie has a goal of jogging 100 miles. The distance she runs each day is the same unit fraction.  What are some possible fractions of a mile she can run each day and the number of days will it take her to reach her goal? Explain how you found your answers” (5.NF.7c).
• Lesson 11.4, On Your Own, Problem 13, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. “Kecia cuts a $$\frac{1}{4}$$ pound pepper into 6 equal-sized pieces. How much of one whole pound is each piece? Represent the problem on the number line (number line with range 0 to 1 given, no intervals labeled).” (5.NF.7c)
• Lesson 11.4, On Your Own, Problem 9, students solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions. For the equation $$\frac{1}{4} ÷ 6 = t$$, Lena writes the word problem ‘A string is 6 feet long. Jen wants to cut the string into 1/4-foot pieces. How many pieces will Jen get?” Why does the word problem not make sense for this equation?’” (5.NF.7c)
• Lesson 15.6, On Your Own, Problem 3, students find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, properties of operations, and/or the relationship between multiplication and division. “A giraffe can cover one mile in about 6.4 minutes. An elephant can cover one mile in about 8.2 minutes. About how much longer will it take the elephant to cover 3 miles than the giraffe?” (5.NBT.6 and 5.NBT.7)
• Lesson 17.6, On Your Own, Problem 6, states “Carlos sells coupon booklets for $5.50 apiece. He makes $60.50. Monica sells the same booklets for $4.75 each and makes $57. Who sells more booklets? How many more?” (5.NBT.7)

Each Unit has a Performance Task that involves real world applications of the mathematics from that unit. For example, the Unit 4 performance task is called “Trail Teamwork” and addresses 5.NF.3, 5.MD.1, and 5.MD.2 as students complete the following:

• Determine what fraction of a 10 mile long hiking trail each of 4 people are in charge of cleaning.
• Determine the distance between equidistant signs along a trail (3 signs within $$\frac{1}{2}$$ mile).
• Determine how many trees are planted if every $$\frac{1}{4}$$ mile a tree is planted. (5.NF.7c) 
• Create line plots to show the heights of those trees after a few weeks.

The instructional materials for HMH Into Math Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. In general, two, or all three, of the aspects are interwoven throughout each module. On the Module planning page, the progression is included in a diagram showing the first few lessons focused on understanding and connecting concepts and skills and the last lessons focused on applying and practicing. 

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

• Lesson 16.1, Check Understanding, Problem 2, students develop conceptual understanding of multiplication with decimals using hundredths grids. “Kali has a banner with a width of 0.3 meter and length of 1.5 meters. What is the area of the banner? Use the decimal model.” (5.NBT.7)
• Lesson 5.5, On Your Own, Problem 16, students use the formula for volume to find the volume of rectangular prisms. Given a cube with the following information: length: 7 cm; width 4 cm, volume 168 cu cm, students “Find the unknown number. Height _ cm.” (5.MD.5b)
• Lesson 3.4, On Your Own, Problem 5,  students solve an application problem. For example, “A local group raised $3,273 during a recent event. The money raised will be shared equally among 3 different charities. How much money will each charity receive? Write an equation to model the situation. Then solve.”  (5.NBT.6)

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

• In Lesson 10.4, Build Understanding, Problem 1, states "The nature preserve has a 3-mile long trail for birdwatchers. The ranger divides the birdwatcher trail into $$\frac{1}{2}$$-mile sections and names each section after a different bird. How many of these sections does the trail have? A. Complete to describe the situation and model it with an expression. 'The trail is ___ miles long and is divided into ____-mile sections. This can be modeled by the expression ___”. 
• In Lesson 11.1, More Practice and Homework, Problem 1, students represent the situation for each problem with a visual model. They write a division equation and a related multiplication equation. “Marcos has 4 gallons of gasoline for his lawn mower. How many lawns can he mow if each lawn uses $$\frac{1}{4}$$ gallon of gasoline?” Students engage in application and conceptual understanding as they complete the problem.  
• In Lesson 11.3, On Your Own, Problem 3  students use application and conceptual understanding to solve “Write and solve a division word problem for the visual model.” The visual model shows five circles partitioned into six pieces each. (5.NF.7c)

The instructional materials reviewed for HMH Into Math Grade 5 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

The MPs are identified at the unit, module and lesson level. In addition, the information in the Planning and Pacing Guide also include references to the MPs. For example:

• Each lesson has a focus Standard for Mathematical Practice (SMP) based on the identified lesson’s learning goal and the tasks that meet the learning goal. These are listed under Lesson Focus and Coherence at the start of each lesson. Additionally, on page 15 of the Planning and Pacing Guide, it outlines for teachers where to look for each of the SMPs. It states, “MP1, MP3, and MP5 are paired with Spark Your Learning tasks. When students connect understanding they have developed with more efficient procedures, MP7 and MP8 are being attended to. This helps students explain and justify their procedures with MP4, MP2 and MP6 are attended to within lessons that ask students to apply procedures in practice.”
• All MPs are clearly identified throughout the materials. For example, MP1 is identified in the Spark Your Learning tasks. MP2 is identified in Lesson 10.5, and Lesson 9.2; MP3 in Lesson 11.1 and Lesson 16.2; MP4 in Lesson 12.1 and Lesson 15.2; MP5 in Lesson 11.1 and Lesson 6.1; MP6 in Lesson 10.5 and Lesson 1.3; MP7 in Lesson 9.2 and Lesson 1.1; and MP8 in Lesson 1.3 and Lesson 15.1.
• The Planning and Pacing Guide for the teachers has a section identified as Correlations for Mathematical Practices. In this section, the 8 Mathematical Practices are listed in a table with a detailed description (from the common core documents) of the practice as well as “some examples” of where the practice is included in the text series. Each math practice has 11 -14 locations listed of where the teachers can look for specific MPs.
• Planning and Pacing Guide pages 17-19 provide additional detail and clarity about each Mathematical Practice. These pages also include Questions to Ask with each Mathematical Practice. These questions provide support to advance students. 
• The Module Review includes a labeled question “Use Tools” in the Student Edition that asks students to choose a tool and explain their choice. The Teacher Edition recommends and provides additional support to have students discuss and share strategies and tools used as part of their review of the module.
• In the Teacher Edition, in the margin under Homework & Test Prep, there is a section that describes MPs that can be seen within the HW worksheet for the students.
• The Teacher Edition for the Spark Your Learning section includes an Advancing or Assessing question to have students choose strategies and tools. Additionally, Build Understanding”recommends to teachers to have students share strategies and tools to solve the problem. 
• Correlations in the Planning and Pacing Guide include lesson examples and to correct over identification. This can be found in the Teacher’s Edition, both on the introductory pages for each lesson and on the lesson pages with Build Understanding and Step It Out Tasks.

However, the materials over-identify the Math Practices, with some identified for every lesson. In addition, some MPs are incorrectly identified. For example: 

• MP1 and MP3 are identified as “in every lesson.”
• MP2 is identified “in every Spark Your Learning Lesson and in most lessons”.
• MP4, MP5, MP6, MP7, and MP8 are identified “in most lessons”. According to the correlations chart, MP4 (Model with Mathematics) can be found in most lessons.  Examples include Lessons 2.1, 3.4, 4.4, etc.
• Lesson 8.6 indicates a focus on MP2 and MP3. However, in the planning it states that MP5 and MP6 are the focus.
• In Modules 6 through 11, MP1 is not explicitly identified as a part of any lesson.  
• In the Planning and Pacing Guide it is indicated that Lesson 6.2 addresses MP1. However, in the Lesson Plan in the Teacher Edition it is not indicated. 
• MP1 is labeled in the correlations chart as in every lesson, but there are no explicit connections in Lessons 1.3, 9.1 or 15.6. 
• Lesson 2.3 lists MP2 and MP3 in the Teacher Edition, Lesson Focus and Coherence section.  Problem 3, in the On Your Own section, is titled Construct Arguments and is identified as such in the Teacher Edition but is not linked to the MP. Problem 2 in the More Practice/Homework page is identified as Construct Arguments.  The Teacher Edition shows how this problem attendants to MP3. It states, “For Problem 2, students will judge the reasonableness of a response using two different sets of compatible numbers.” 
• Multiple problems within a lesson include MP language with no direct connection to MPs. For example, Lesson 5.2, Problem 4, states Attend to Precision prior to listing the question. Lesson 15.5, Problem 12,  states Critique Reasoning. These MP phrases are in all modules and are in bold prior to the question or problem being posed.

For the most part, when identified, MPs are used to enrich the mathematical content of the lessons. For example:

• Lesson 4.2, Problem 1 identifies MP3. It explains students evaluate Rafiq’s reasoning in comparing two expressions and are able to explain whether or not Rafiq made an error. 
• Lesson 15.1, More Practice/Homework, identifies MP2 in Problem 13, as students extend their reasoning of multiplication patterns with decimals to higher order powers of 10. 

The instructional materials reviewed HMH Into Math Grade 5 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. 

The materials do not attend to the full meaning of MP4 and MP5. 

Students have limited opportunity to engage with MP4 to model with mathematics. Models are often provided for the students. For example: 

• Lesson 9.4, Step It Out, students write an equation. It states “Toni has a plaque that is $$5\frac{1}{2}$$ inches wide and $$8\frac{1}{2}$$ inches long. Toni hangs her plaque on a wall. How much wall space does the plaque cover?”
• In Lesson 7.6, Step it Out, students draw bar models to represent the problem. 
• In Lesson 9.4, Step it Out, it states “Write and solve an equation to model the problem using fractions greater than 1.”
• In Lesson 10.4, On Your Own, students solve whole number by unit fraction division problems and they are prompted to draw visual models to complete and investigate the equations. 

Examples of MP5 being identified, where students do not choose tools strategically, as the tools are given to students include:

• In Lesson 14.1, Build Understanding, students are directed on which tool to use. It states “What base-ten blocks do you use to show 0.36 and 0.15?”   
• In Lesson 14.4, On Your Own, Problem 6, students are directed on which tool to use. It states,“Use the addition chart to find the price.”
• In Lesson 14.3, On Your Own, Problem 5, students are directed on which tool to use. It states“Justify your answer using the number line.”

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

• MP1: In Lesson 7.1, Spark Your Learning, states “Ms. Fong mixes amounts of water, glue, and laundry detergent together to make slime. Each amount is a fraction of a liter. Use a visual model to estimate the total number of liters of ingredients she mixes together.” Persevere, the Teacher Edition states “If student needs support, guide them by asking: ‘What do you need to find? What about these fractions makes them difficult to add? How could you use fraction strips to estimate the sum?’”
• MP2: In Lesson 14.5, Step It Out, Problem 1, students reason abstractly and quantitatively to answer, “Santiago is at the store with his brother. He reads the total for his purchase on the cash register and realizes that he has only $4.73. His brother pays the difference. How much does Santiago’s brother pay? Write an equation to estimate how much Santiago’s brother pays.”
• 
  MP6: In Lesson 20.2, On Your Own, Problem 7, students learn to attend to precision, “Compare the triangles: Name one attribute that three of the triangles have. Name one attribute that one of the triangles has.”
• MP7: In Lesson 1.1, Build Understanding, students look for and make use of structure to solve,  “Complete each column in the table. How does the number of zeros in each number change?” “In the ‘10 times as much’ column? In the ‘1 times as much’ column? In the ‘1/10 of ‘ column?”
• MP8: In Lesson 17.1, On Your Own, Problem 2, students look for repeated reasoning to solve, “The container holds 100 servings of juice. Show a pattern to find the amount of juice in one serving. How many gallons are there in one serving?”

The instructional materials reviewed for HMH Into Math Grade 5 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Turn and Talk sections often 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, especially in problems labeled “Critique Reasoning”.

• Lesson 4.4, On Your Own, Problem 11,  “Deshawn says that he can evaluate the numerical expression 7 + (3 × 8) – 5 without parentheses and get the same answer. Is Deshawn correct? Explain how you know.”
• Lesson 8.6, On Your Own, Problem 12, “Carl and Maeve are asked to think of a fraction and multiply it by 5,267. Carl thinks of $$\frac{5}{6}$$. Maeve thinks of $$\frac{7}{7}$$. They both say their product is less than 5,267. Are they correct? Explain.”
• Lesson 4.2, Critique Reasoning, Problem 4, “James is making banners for his club’s airplanes to pull. Each banner is 5 feet long and is attached by a 10-foot long rope. He models the total length of the banners and rope for six airplanes with the numerical expression 6 × (5 + 10). He says the total length for six planes is five times as great as the total length needed for one plane. Correct his error.” 
• Lesson 8.6, Spark Your Learning, “The painting shown is resized to 3.4 of its original size. How does the height of the resized painting compare to the height of the original painting? Is the height of the resized more than or less than $$\frac{3}{4}$$ foot? Draw a visual model to represent your thinking. Justify your reasoning.” 
• Lesson 15.5, Turn and Talk, “Is your answer reasonable? Explain.”  
• Lesson 11.5, Turn and Talk, “Why should you divide each half of the rectangle into 4 equal groups?” and “Does it matter what visual model you use to find the quotient of a unit fraction divided by a whole number? Why might you choose one model over another? Explain.”

The instructional materials reviewed for HMH Into Math Grade 5 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, with teacher prompts to promote explaining their reasoning to each other. Independent problems provided throughout the lessons also have teacher guidance to assist teachers in engaging students. Examples include the following:

• The Teacher Edition provides Guided Student Discussion with guiding questions for teachers to create opportunities for students to engage in mathematical discourse. Module 14, “How can you tell that the 4-digit number at the top of the subtraction problem is less than 2,000?” Module 19 “What would have happened if the instruction in step D had the pattern start at 95 instead of 83?”
• Critique, Correct, and Clarify, is a strategy used to assist students in constructing viable arguments. Lesson 1.4, On Your Own, “Have students work out the steps to multiply on their own. Encourage students to describe the error and review explanations with a partner. Students should refine their responses after their discussions with a partner.” Lesson 5.4, On Your Own, Problem 12, “Point out to students that Problem 12 can be solved more than one way. As shown, the volume of the new cube can be multiplied by the number of cubes: $$8 \times (2 \times 3 \times 4)$$. Or the length of each side of the cube can be doubled: $$6 \times 4 \times 8$$. Encourage students to describe different ways of solving the problem with their partners. Students should refine responses after their discussions.”
• In Lesson 2.4, students tell if an estimate is reasonable and explain why. Teacher guidance states, “Problem 3, Construct Arguments shows that students need to determine the reasonableness of a quotient.”
• The Teacher Edition includes Turn and Talk in margin notes to prompt student engagement. Lesson 11.1, Build Understanding, “Have students share their reasoning. For students who are struggling, suggest that they compare the multiplication equation with their visual models.”  The Turn and Talk builds off of earlier discussion questions such as “What numerical information is given in the problem? How can you show the number of pounds of potato salad? How can you show the $$\frac{1}{4}$$-pound servings in your visual model?"

The instructional materials reviewed for HMH Into Math Grade 5 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 and accurate terminology and definitions when describing mathematics and support students in using them. Examples are found throughout the materials. 

• At the beginning of each module, Key Academic Vocabulary is highlighted for the teacher.  These sections include both Prior Learning. Review Vocabulary and Current Development and New Vocabulary. Definitions are given for each vocabulary word. 
• Within the Student pages, new vocabulary is introduced in highlighted sections called Connect to Vocabulary. Lesson 4.1: “You can model a context mathematically using a numerical expression. A numerical expression is a mathematical phrase that uses only numbers and operation signs”. In Lesson 20.3 it states “There are two accepted definitions of a trapezoid. One definition defines a trapezoid as having exactly one pair of parallel sides. The other definition defines a trapezoid as having at least one pair of parallel sides.”
• In the module planning pages, there is a Linguistic Note on the Language Development page that provides teachers with possible misconceptions relating to academic language. Unit 1: “Many English words have multiple meanings that can interfere with comprehension. For example, flat and long are typically used as adjectives; however, they also name the base-ten blocks used for understanding multi-digit place values.”
• In the lesson planning pages, Sharpen Skills in some lessons include Vocabulary Review activities. Lesson 20.1:  “Objective: Students review types of polygons. Have students work in types of polygons. List the following review terms on the board: triangle, decagon, hexagon, octagon, quadrilateral. Ask students to discuss what attributes all of these figures have in common. Then have students identify the specific characteristics of each. Have students form pairs and each student should draw an example of each figure listed. Students should compare their figures.”
• Guide Student Discussion provides prompts related to understanding vocabulary. Module 6: “Listen for students who correctly use vocabulary as part of their discourse. Students should be familiar with the terms fraction, whole, numerator, denominator, and equal parts. Ask students to explain what they mean if they use those terms. How can you tell how many large triangles will fit in the puzzle? How can you use this fact to write a fraction for each large triangle? How can you use these facts to find the fractional area of each small triangle?”
• Vocabulary is highlighted and italicized within each lesson in the materials. 
• There is a vocabulary review at the end of each module. Students do fill-in-the-blank with definitions or examples, create graphic organizers to help make sense of terms, or the teacher is prompted to make an Anchor Chart where students define terms with words and pictures, trying to make connections among concepts. 

The instructional materials for HMH Into Math Grade 5 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.

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

Check Understanding and On Your Own sections include exercises that ask students to use the newly learned mathematics in each lesson. Additional Practice and Homework is available in separate student journal, providing more exercises for students to solve.

For example in Lesson 6.4, Spark Your Learning, students experiment with the following problem: “TV show hosts, Ben and Isa, are preparing small pieces of French toast from slices of bread. Ben cuts his slice into two equal sections. Isa cuts her slice into three equal sections. They were supposed to all be cut into the same-size pieces. Draw a visual model to show what Ben and Isa should do to correct the sizes of their sections of French toast.” Students develop their understanding in the Build Understanding and Step It Out where they write common multiples and equivalent fractions with a common denominator. Then in the On Your Own section students use common denominators to practice writing equivalent fractions. 

The instructional materials for HMH Into Math Grade 4 meet the expectations that the design of assignments is intentional.

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

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

For example, in Lesson 1.2, students begin by using manipulatives or a visual model to solve the following problem: “Aaron’s school has set up a website to raise money for a new playground. Each donation is $10. For the first 10 days, the school receives 10 donations each day. How much does the school received the first day? How much does the school receive after 10 days? Explain your thinking.” Students then look for patterns in Step it Out first by multiplying 7 x 10 then 7 x 10 x 10 etc. In On Your Own, they progress to the abstract concept of 5 x $$10^3$$.

Concepts are developed throughout the modules such as 5.NBT.2.7 which is introduced in Module 14, Add and Subtract Decimals. Students continue work with decimals in Module 15, Multiply Decimals and Whole Number, Module 16 Multiply Decimals, and Module 17 Divide Decimals.

The instructional materials for Into Math Grade 5 meet the expectations for having a variety in what students are asked to produce. For example:

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

Homework assignments ask for a variety of responses from fluency to higher-level thinking. For example, Lesson 4.4, Homework has seven problems. One problem is a multi step word problem. One asks students to describe how they would solve {[24 / (6 - 2)] - 4} x 3. Four problems ask students to use parentheses to make an expression equal to a given value and the last problem asks students to critique the reasoning of another students work that is provided. 

The instructional materials reviewed for HMH Into Math Grade 5 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

At the beginning of the lesson they indicate what materials/manipulatives will be needed. The lesson on the student pages provide a picture of the manipulative that they will use. For example, in Lesson 4.1, students are asked to model a situation with two color counters “Al drum line is made up of 14 fourth-grade drummers and 12 fifth-grade drummers. The fourth-grade drummers stand in a line, and the firth grade-drummers stand in a line behind them.” Students use two color counters or a visual model to create a representation. The manipulatives provide opportunities for students to develop a conceptual model of problems that they will then represent in pictorial form in their student workbook. 

Lesson 6.3, Spark Your Learning, students are encouraged to use a visual representation to show $$\frac{3}{4}$$ and $$\frac{1}{2}$$. They use fractions tiles to make equivalent fractions in order to subtract the fractions. The work with fraction tiles is connected to the written method for making equivalent fractions and subtracting fractions.

Examples of manipulatives for Grade 5 include: base ten blocks, connecting cubes, fraction strips, grid paper, ruler, number line, square tiles, and two color counters

The materials rely heavily on pictures of manipulatives. When physical manipulatives are called for in the lesson materials in the Teacher Edition, it is not always clear how they are to be used. There is sometimes direction for how they can be used in the differentiation section.

The visual design of HMH Into Math Grade 5 is not distracting or chaotic.  The materials, both print and digital, follow 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. There is much information provided, and 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. The teacher edition provides information for teachers to be able to access digital resources. There is room for students to record answers and show their thinking.

Features of the materials are consistently presented, and the use of colored fonts supports identification of lesson components. For example, Turn and Talk opportunities are highlighted in yellow and Check for Understanding is always in red. Visual images mirror the situation in the problem or can be used by students as they solve the problem. For example, Module 5, Lesson 5, Step it Out, includes a drawing of a fish tank on the side of the problem. Students use the dimensions draw on the fish tank to write a formula for the volume.

The instructional materials for HMH Into Math Grade 5 meet the expectations for providing quality questions to help guide students’ mathematical development. 

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

• Lesson 4.2, Activate Prior Knowledge, provides teachers with following questions: “Where did you put the parentheses? Why? Which operation will you do first to evaluate the expression? What should happen in your word problem when the expression shows addition? What should happen in your word problem when the expression shows division?” 
• Lesson 19.4, Step it Out, includes questions for teachers to ask students: “How will you complete Antonia’s row in the table? How will you complete Connor’s row in the table? ” “How does writing the numbers as ordered pairs help you analyze their relationship?” 
• Lesson 13.3, Spark Your Learning, includes four questions: “How do you round a whole number, like 853, to the nearest ten?” “How does the value of the number 15.04 compare to the value of the number 15.040?” ”Which digits in the numbers 15.042 and 15.046 have the place value of hundredths?” 

The instructional materials for HMH Into Math Grade 5 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.

Throughout the teacher edition there are ample and useful annotations and suggestions on how to present the content in the student edition to all levels of math learners in the classroom. Specifically, the teacher edition includes sections titled, Lesson Focus and Coherence, Plan for Differentiated Instruction, Spark Your Learning, Motivate, Persevere, and Leveled Questions, designed to assist the teacher in the mathematical standards and concepts that are present in the lesson. For example: 

• Lesson 2.2, Spark Your Learning, includes margin notes for the teacher that  describes possible strategies the teacher can take based on the needs of the students.  
• Lesson 7.1, Activate Prior Knowledge, include the following guidance for teachers:  “Based on the students’ responses to the Problem of the Day, choose one of the following: Project interactive Reteach, Grade 4, Lesson 11.2 or Complete the Prerequisite Skills Activity.”

The instructional materials for HMH Into Math Grade 5 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 that teachers can improve their own knowledge of the subject. Examples of the grade-level explanations include:

This material includes explanations and examples of the course-level mathematics specifically for teachers that can improve their own knowledge of the subject. In the Teacher Edition modules there are small sections of examples and support for the adult in the math classroom as it relates to math standards. For example: 

• The Mathematical Progressions table in each module and lesson highlights Prior Learning, Current Development and Future Connections. In Lesson 14.2 this table lists the 4th grade standard that supports the 5th grade on-level standard and what 6th grade standard this will lead into. 
• Professional Learning notes are present in each lesson. In Lesson 14.2, Professional Learning, discusses “Using Mathematical Practices and Processes: This lesson provides students with an opportunity to use base-ten blocks and draw quick pictures to subtract decimals. Like they did with the previous tools in adding decimals, students adapt the tools to subtract decimals like they did with whole numbers. Developing proficiency with these tools will enable students to understand the principles of using place-value strategies to regroup when subtracting.” 
• The Planning and Pacing book provided in the series includes a correlation chart for the math practices that only defines each math practice in full.

The instructional materials for HMH Into Math Grade 5 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 which lists prior learning, current development, and future connections. Similarly, the beginning of each lesson in the Teacher Edition includes Mathematical Progressions that show connections to prior and future grades’ standards, as well as other lessons within the program.

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

The instructional materials for HMH Into Math Grade 5 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 that includes the standards and pacing (number of days) for each lesson. There is another standards chart in the Planning and Pacing Guide that lists each standard and correlation to Student Edition Lessons. In the Teacher Edition, the standards contained in each lesson are identified with written descriptions as well as listed under Current Development in the Mathematical Progressions chart.

 The instructional materials for HMH Into Math Grade 5 include strategies for parents to support their students progress. The Family Resources tab include several resources for parents:

• “School Home 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. This letter is available in English, Spanish, Haitian-Creole, and Portuguese.
• Math on the Spot videos are available for specific lessons within a module online in Family Resources. For example, Module 13 includes a Math on the Spot video for Lessons 1, 3, and 4.

The instructional materials for Into Math Grade 5 explain instructional approaches used and how they are research-based.

The Planning and Pacing Guide contains Teacher Support Pages that include a section on Supporting Best Practices. “Into Math classrooms maximize student growth by providing teachers with content 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) that “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 that “help teachers promote the design principles during instruction.” Each module contains a Language Development page in the Teacher Edition that states where the language routines should be used. On the lesson pages of the Teacher Edition, there are Support-Sense Making boxes that describe how the language routine can be used. Also, there are notes in the margin of the teacher’s edition providing connections from the strategy to the principle.

The instructional materials for HMH Into Math Grade 5 meet the expectations for providing strategies for gathering information about students’ prior knowledge within and across grade levels.

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

The instructional materials for HMH Into Math Grade 5 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 that identifies where students may make a mistake or exhibit misunderstanding. There is a rationale that explains the likely misunderstanding and suggests instructional adjustments or steps to help address the misconceptions.
• There are also “Watch For” boxes and question prompts that highlight areas of potential student misconceptions.

The instructional materials for HMH Into Math Grade 5 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 a few Spiral Review questions for ongoing practice in the More Practice/Homework section.
• Online interactive lessons and homework practice provide students with immediate notification that answers are correct or incorrect, but do not provide feedback for changing incorrect answers.
• The online lessons are the same as in the print textbook.
• 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. The online Module Review does provide “Hints” to help with selection of answers.
• Digital assessments are auto-scored and generate recommendations that can provide feedback to teachers, but not directly to students.

The materials provide support for ongoing review and practice. Within each lesson there is an Activate Prior Knowledge/Problem of the Day. This is a review problem from prior units/lessons. Directly correlated with each question, there is teacher support on next steps based on the students’ responses.

The instructional materials for HMH Into Math Grade 5 meet the expectations that assessments clearly denote which standards are being emphasized.

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

Each Unit has a summative Performance Task that includes the content focus in the teacher pages of the Assessment Guide.

The instructional materials for HMH Into Math Grade 5 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 that when using Online Ed, teachers can assign the Are You Ready digitally “to power actionable reports including, proficiency by standards, and item analysis.”
• “Check Understanding is a quick formative assessment in every lesson. Teachers use data to determine which students need additional small-group support and which students can continue on to independent practice or math center challenges.” (Planning and Pacing Guide) Check Understanding presents a limited number of questions, usually 1-3, which includes a digital option that can be assigned “to determine success with the learning objective, items to review, grouping and differentiation resources.”
• Each performance task includes a task-specific rubric indicating a level 0 response through a level 3 response. The structure of the rubrics is the same, but specific words are changed to reflect the mathematical content of the module. Level 3 indicates that the student made sense of the task, has complete and correct answers, and checked their work or provided full explanations. Level 2 indicates that the student made sense of the problem, made minor errors in computation or didn’t fully explain answers. Level 1 indicates that the students made sense of some components of the task but had significant errors in the process. Level 0 shows little evidence that the student has made sense of the task or addressed any expected components and has an inability to complete the processes.
• The Individual Record Forms in the Assessment Guide suggest Reteach Lessons that 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 that teachers can use for follow-up with students.
• The Performance Task Rubrics for the Unit Performance Tasks do suggest Reteach Lessons but do not provide any other guidance that teachers can use for follow-up with students.

The instructional materials for HMH Into Math Grade 5 include Scales to Track Learning Goals at the end of each lesson. The Teacher Edition introduction states, “The scale below can help you and your students understand their progress on a learning goal. ”

Each lesson contains “I Can” scales with four levels of  “I Can” statements written in increased difficulty. While there is a note stating, “The scale below can help you and your students understand their progress on a learning goal,”  there is no explicit indication of how to use these scales. 

At the end of On Your Own section is a Learning Mindset prompt where students write a response to reflect on the lesson. For example from Lesson 5.1 the Learning Mindset asks “What is still unclear about unit cubes?”

The instructional materials for HMH Into Math Grade 5 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 makes connections to both prior and future skills and standards to scaffold instruction.
• At the beginning of each module, Diagnostic Assessment, Are You Ready?, allows teachers to “diagnose prerequisite mastery, identify intervention needs, and modify or set up leveled groups.”
• Each lesson provides Warm-up Options to activate prior knowledge such as Problem of the Day, Quick Check for Homework, and Make Connections.
• Throughout the lessons, there are notes, strategies, sample guided discussion questions, and possible misconceptions that provide teachers structure in making content accessible to all learners.
• Student practice starts with up to four Check Understanding exercises to complete with guidance before moving to independent work in On Your Own or More Practice/Homework.

The instructional materials for HMH Into Math Grade 5 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

• There are Reteach and Challenge activities for each lesson.
• Each module includes Plan for Differentiated Instruction that provides teachers with teacher-guided, Small-Group Options and self-directed Math Center Options based on student need: On Track, Almost There (RtI), and Ready for More.
• Each lesson provides Leveled Questions in the teacher’s edition identified as DOK 1, 2, and 3 with an explanation of the knowledge those questions uncover about student understanding.
• There are four 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.” 3) “Compare and Connect: Students listen to a partner’s solution strategy and then identify, compare, and contrast this mathematical strategy. And 4) “Critique, Correct, and Clarify: Students correct work that is not their own with a flawed explanation, argument, or solution method and share with a partner to reflect and then refine the sample work.”

The instructional materials for HMH Into Math Grade 5 partially meets the expectations for embedding tasks with multiple entry points that can be solved using a variety of solution strategies or representations.

The Planning and Pacing Guide, Teacher Support, Access and Equity, and Spark Your Learning Tasks are “designed as ‘low-floor/high ceiling’ tasks that all students can access but that can also be extended to provide challenge.”  Teachers are provided guidance on how to assist various levels of learners, depending on how they respond to the problem. For example, Lesson 15.2, Spark Your Learning has this prompt: “Mary is shopping at the grocery store. She finds the weight of one banana and buys two more bananas that weigh the same amount.   What is the weight of 3 bananas? Justify your thinking.” A picture of a scale shows the banana weighs 0.28 pound. This problem has multiple entry points and solution methods. However, Spark you Learning is not present in every lesson. 

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

The instructional materials for HMH Into Math Grade 5 meet the expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

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

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

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

The instructional materials for HMH Into Math Grade 5 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:

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

The instructional materials for HMH Into Math Grade 5 meet the expectations for providing a balanced portrayal of various demographic and personal characteristics.

The materials provide various demographic and personal characteristics. The instructional materials reviewed for this series meet the expectation for providing a balanced portrayal of various demographic and personal characteristics. Pictures of adults and children in the text show a variety of demographics and personal characteristics

There are a variety of names used in word problems throughout the text. The lessons contain a variety of tasks and situations in the story problems that interest students of various demographic and personal characteristics. There is a balanced approach to the use of gender identification.

The instructional materials reviewed for HMH Into Math Grade 5 provide opportunities for teachers to use a variety of grouping strategies.

In the Planning and Pacing Guide there is a section titled, Grouping and Recommendations. This section states, “One of the most valuable and time-saving tools for teachers is the Recommend Groups tool online. It synthesizes data from assessments and places students into leveled groups, which teacher can modify as needed. Recommended lesson-level resources for each group surfaced in the tool and can quickly be assigned to each group.”

• 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, there are ample opportunities 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.

The instructional materials reviewed for HMH Into Math Grade 5 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.

The instructional materials reviewed for HMH Into Math Grade 5 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 bar diagrams, drag and drop the correct values into a table, or graph an equation.
• iTools virtual manipulatives are available online.
• 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.

The instructional materials reviewed for HMH Into Math Grade 5 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 that connect to the internet with an applicable browser.

The instructional materials reviewed for HMH Into Math Grade 5 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.

• 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.
• There are growth monitoring assessments that 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 that repeats the questions presented throughout the interactive lessons. However, teachers can modify questions and 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.

The instructional materials reviewed for HMH Into Math Grade 5 are intended to include opportunities for teachers to personalize learning for all students.

• 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 Grade 5 provide minimal opportunity to be adapted for local use.

• Pieces of a lesson can be assigned directly to students or groups of students.
• There is a question bank for teachers to create assessments. The bank repeats the questions that are already included in each lesson, and these questions can be modified.

The instructional materials reviewed for HMH Into Math Grade 5 do not incorporate technology that provides opportunities for multiple students to collaborate with the teacher or one another.