The instructional materials for HMH Into Math Grade 4 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 2, Performance Task, Question 2, “Fei is taking a 5,286-mile road trip with two other friends. If they split the drive equally, how many miles will each person drive?” (4.NBT.6)
• Module 12, Form A, Question 6, students determine if a given fraction with a denominator of 100 is equivalent to a given decimal. They are given “$$\frac{9}{100}$$ and 0.9,  $$\frac{40}{100}$$ and 0.40, and $$\frac{80}{100}$$ and .80”. (4.NF.6)
• Module 14, Form A, Question 5, students solve a story problem requiring addition and subtraction of fractions with like denominators. (4.NF.3d)
• Module 19, Form A, Question 9, “Marcus buys an 8-pound pumpkin. He takes it home and removes 12 ounces of seeds and pulp. How many ounces does his pumpkin weigh now?” (4.MD.2)
• End-of-Year-Test, Question 7, “Jamal's class collects 37 board games. Chen's class collects 3 times as many as Jamal's class. The classes donate all the games to 9 local groups. Each group receives about the same number of games. Which is the BEST estimate of how many games each group receives? Students choose from 4, 16, 28 and 32”. (4.OA.3)
• For Module 20, Form A, Question 10, students convert from centimeters to millimeters instead of from millimeters to centimeters requiring students to convert from larger units to smaller units. (4.MD.1) Question 13, students convert from liters to milliliters. (4.MD.1)

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:

• Modules 11 and 13, the following problems use denominators outside of the range of possible denominators for 4.NF: Module 11, Form B, Questions 2 and 3, and Module 13, Form A, Question 7.

The instructional materials reviewed for HMH Into Math Grade 4 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 13 out of 21, which is approximately 62%.
• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 70 out of 108, which is approximately 65%.
• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 107 out of 174 days, which is approximately 61%.

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 65% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for HMH Into Math Grade 4 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:

• Lessons 10.1- 10.4, 4.OA.4 supports the major work of 4.NBT.2. Lesson 2, Question 7, students decide if 7 is a factor of 91 and justify their answer by using division. Lesson 3, Question 4, students solve a story problem by using 4 as a factor. 
• Lesson 11.4, connections are made between the major work of 4.NF.1 and the supporting work of 4.OA.4. In Question 2, students list the factors of 8 and 12, referring to the fraction 8/12. Students use common factors between 8 and 12 to write equivalent fractions. 
• Lesson 19.5, 4.MD.4 supports the major work of 4.NF.2. Question 1, students plot fractions on a line plot requiring students to first create equivalent fractions, then compare and order them correctly along the line plot.
• Lesson 19.5, On Your Own, Question 7, 4.MD.4 supports the major work of 4.NF.3d when students solve fraction addition and subtraction problems with data given in line plots. “A wildlife center records the weights of six reptiles. The fractions (all with a denominator of 8) represent the weights, in pounds, of the reptiles. Make a line plot to display the data. What is the difference in the weight between the heaviest and lightest reptiles weighed?”

The instructional materials for  HMH Into Math Grade 4 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 174 days, 123 days for lessons and 52 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 4 has 7 units with 21 modules that contain 108 lessons. 
• The Planning and Pacing guide designates 8 lessons as 2-day lessons and 100 as 1-day lessons, leading to a total of 116 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 7 unit openers for Grade 4 (7 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 123 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 7 Performance Task for Grade 4 (7 days). 
• Each module has both a review and an assessment. There are 21 modules (42 days). 
• Based on this, 52 assessment days can be added.

The instructional materials for HMH Into Math Grade 4 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, Module18, Lesson 2, Prior Learning, builds upon work done during Grade 2 where students identified and drew two-dimensional shapes. Future Connections notes the work will continue in Grade 5, Module 20, Lesson 1, where students will identify and classify polygons. Additional features of the materials further support the progressions of the Standards. These include:

• In the beginning of each module there is a diagnostic assessment Are You Ready? that identifies prior knowledge needed for the current module. Module 5 shows the link to prior learning for Multiplication Facts as Grade 3, Module 4 and 5 in the Data-Driven Intervention Chart. A narrative is provided for each skill on the page Multiplication Facts. These items assess whether students are able to find the product of two 1-digit numbers using a variety of strategies. In upcoming lessons, students may use these strategies to multiply larger numbers.”
• In each lesson the standard of focus is explicitly connected to work in future and prior grades.  For instance, Module 20, Lesson 2, identifies the lesson focus as “Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.” (4.MD.1) 

There is one instance of off-grade level work that is not clearly marked:

• Lesson 13.4, Task 2 states  “How could you figure out what angle measure is $$\frac{1}{4}$$ of a circle?”  The answer is “Divide 360 degrees by 4”. (4.NF.4)

The materials give students extensive work with grade-level problems. 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 8.3, Build Understanding, students relate area models to partial products to multiply two-digit by two-digit numbers. For example, Question 1B,  “Show how you can use an area model to represent the problem?” Question 1C, “How can you write multiplication sentences to find the partial products?” During On Your Own, students solve Question 2: “Complete the area model. Write and solve an equation for the area model.” Additional practice is provided in More Practice/Homework. (4.NBT.5)
• Lesson 12.4, Step It Out, students compare decimals to hundredths using hundredths grids, number lines, and place value charts. During On Your Own, students solve Question 3:  “Chris has two kittens, Oscar and Tiger. Which kitten is heavier? Shade the hundredths model for each weight. Locate and label the weights on the number line." (4.NF.7)
• Lesson 19.5, Step it Out, students represent and interpret measurement data in line plots. Question 1:  “The weights of some cell phones and tablets are shown. How can you display the data using a line plot?” The On Your Own section has 6 questions where students create line plots and interpret the data. Question 7: “A local pizzeria held a pizza-eating contest. The fractions below represent the amount of pizza each contestant ate in 5 minutes.  Make a line plot to display the data. How much more pizza did the winner eat than the person who came in last place?” (4.MD.4)

The materials relate grade level concepts to prior knowledge from earlier grades. 

• 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.
• Lesson 1.4, Compare and Order Numbers students build upon prior learning whereby they “understood that the three digits of a three-digit number represent amounts of hundreds, tens, and ones Grade2, Lessons 6.4 and 6.5 ”. The lesson additionally includes a Make Connections section where it suggests Project the Interactive Reteach, Grade 2, Lesson 6.4. And, Complete the Prerequisite Skills Activity where a problem is presented to the students. Students explain how they can compare numbers and how they can place numbers in order from least to greatest.
• Lesson 6.2, Investigate Remainders students build upon their prior knowledge of division by separating objects into equal groups (3.OA.2) and  by representing division using arrays and bar models. (3.OA.3) Students apply this knowledge to investigate remainders. (4.NBT.6) For example, Module 6, Lesson 2, Question 2, students solve a word problem where 35 pencils are needed to make 8 party favor bags. Students are prompted to create a drawing to show how all the pencils can be divided. Then, describe how many total pencils were drawn, how many pencils went into each bag, and how many pencils were left over (remainder) and to tell why.
• Lesson 11.3, Explain Fraction Equivalence Using Visual Models, students to build upon their prior learning where they “recognized equivalent fractions as the same size or point on a number line Grade 3, Lessons 16.1 and 16.2;  generated simple equivalent fractions, Grade 3, Lesson 16.3; and, expressed whole numbers as fractions (Grade 3, Lesson 13.5).” The lesson additionally includes a Make Connections section where it suggests Project the Interactive Reteach, Grade 3, Lesson 16.1. and Complete the Prerequisite Skills Activity where a problem is presented to the students. They are asked to draw a visual model. Next they are to change the visual model to show a fraction that is equivalent to $$\frac{1}{2}$$. Students are asked “to summarize what is the same about all of the equivalent fractions.” 

The instructional materials reviewed for HMH Into Math Grade 4 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 Unit 1 learning objective is “Place Value and Whole Number Operations” which is shaped by the cluster heading Generalizing Place Value Understanding for Multi-Digit Whole Numbers. (4.NBT.1)
• In Lesson 11.7, the learning objective is comparisons to order fractions which is shaped by 4.NF.1: “Extend understanding of fraction equivalence and ordering.”
• In Lesson 1.2 the learning objective is place value relationships to read and write multi-digit whole numbers to 1,000,000 in different forms which is shaped by 4.NBT.1: “Generalize place value understanding for multi-digit whole numbers.”
• In Lesson 12.4, the learning objective is to compare decimals using visual models, number lines, or place value which is shaped by 4.NF.3: “Understand decimal notation for fractions, and compare decimal 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. Examples of this include:

• Lesson 14.6, connects 4.NF.5 with 4.NF.1 when students use equivalent fractions to write fractions with denominators of 10 as denominators of 100 and then add like denominators. For example, On My Own, Question 9: “$$\frac{22}{100} + \frac{7}{10}$$.”
• Lesson 19.4, Question 1, connects 4.MD.1 to 4.OA.1. Students draw a visual representation of how quarts, pints, and cups compare to gallons. They then describe their comparisons in words.
• Lesson 11.3 connects 4.NF.2 to 4.OA.1. For example, Question 2, students create a fraction equivalent to $$\frac{1}{2}$$ with 6 as the denominator. Students are then asked to describe the relationship between the numerator and denominator of each fraction.

The instructional materials for HMH Into Math Grade 4 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 the following:

• Lesson 1.2, Spark Your Learning states “The total area of Texas is two hundred sixty‑eight thousand, five hundred ninety‑six square miles. It is the largest state in the southern U.S. How can you write the total area of Texas in two different ways using numbers?” (4.NBT.2 )
• Lesson 6.4, students use area models and the distributive property to represent division. (4.NBT.5)
• Lessons 7.2, Build Understanding, students use base ten blocks to represent division with one digit divisors. Problem 1 states “Fran has 423 scrapbook stickers. She wants to put an equal number of stickers in 3 different scrapbooks. How many stickers can she put in each scrapbook? Find $$423 \div 3$$.  Use base ten blocks to show the division.” (4.NBT.6)
• Lesson 5.4, students use the distributive property and partial products to multiply 1 digit by 4 digit numbers. (4.NBT.5)
• Module 11, Opening Activity, provides students with four different squares partitioned and shaded differently. In a Turn and Talk they are asked “Which square’s shading represents a different amount? How could you change the shading in that square to make it represent the same amount as the others?” (4.NF.A.2)
• Lesson 11.1, Spark Your Learning states “Liz and Alvin have the same go-karts in different colors. The fuel tank in Liz’s go-kart is $$\frac{3}{5}$$ full. The fuel tank in Alvin’s go-kart is $$\frac{1}{3}$$ full. Whose go-kart has more fuel? How do you know?” (4.NF.2)
• Lesson 11.2, Spark Your Learning states “Abbot and Rowan go to the Climb-a-thon. They both climb ropes that are the same length. Who climbs higher than halfway up the rope?” The problem includes a table representing $$\frac{5}{8}$$ and $$\frac{4}{10}$$ as well as visuals of the same size rope and two students. (4.NF.2) 
• Lesson 11.3, Check Understanding states “Jason makes a $$\frac{5}{6}$$ turn on his skateboard. Samantha makes a $$\frac{10}{12}$$ turn. Did they make the same-sized turn? Use the visual models to explain.” (4.NF.2) 
• Lesson 11.5, Question 4 states “Jerry has two same size circles divided into the same number of equal parts. One circle has $$\frac{3}{4}$$ of the parts shaded and the other has $$\frac{2}{3}$$ of the parts shaded. His sister says that the least number of pieces each circle could be divided into is 7. Is his sister correct? Explain.” (4.NF.2) 
• Lesson 11.6, Question 5 states “Isaiah hikes $$\frac{11}{12}$$ mile along the Lake View Trail. Cheryl hikes 3.5 miles along the same trail. Who hikes farther? Use a fraction comparison strategy to support your reasoning.” (4.NF.2)

The instructional materials for HMH Into Math Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. 

The materials include problems and questions 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 4.NBT.5. Examples include but are not limited to: 

• Module 2 focuses on Addition and Subtraction of Whole Numbers. In Lesson 2.1, On Your Own, students “Estimate. Then find the sum” for problems such as Problem 8: 609,987 + 123,654.” In Lesson 2.2, More Practice, students “Estimate. Then find the difference” for problems 3 -5. “Problem 4: 38,207 - 28,278.” (4.NBT.4)
• Lesson 5.6, On Your Own, Problems 6 - 8, “Estimate. Then write the problem vertically to find the product.” Problem 6 states “$$6 \times 523$$”; Problem 7 states “$$9 \times 5,181$$”; Problem 8 states “$$8 \times 6,719$$.” (4.NBT.5)
• Module 7, Review,  “Divide and Check, Problem 6: $$231 ÷ 5$$; Problem 7: $$458 ÷3$$; Problem 11 $$2,551 ÷ 7$$.” (4.NBT.5)
• Lesson 8.1, On Your Own, students multiply with tens. Examples include “Choose a method. Then find the product.” Problem 5 “$$80 \times 29 =$$ ___”; Problem 6 “$$35 \times 30 =$$ ___”; and Problem 7 “$$90 \times 16 =$$ ___.” (4.NBT.4)
• Lesson 15.3, On Your Own, Problems 6 -9 states “Find the Sum. Write your answer as a mixed number.” Problem 6 “$$1 \frac{9}{10} + 1 \frac{8}{10} =$$ __”; Problem 7 “$$3\frac{1}{4} + 1\frac{2}{4} =$$ ___”; Problem 8, “$$\frac{5}{6} + \frac{5}{6} + \frac{5}{6} =$$ ___”; and Problem 9, “$$2\frac{5}{8} + 1\frac{7}{8} =$$ ___.” (4.NF.3c)
• Additional fluency practice can be found in the More Practice/Homework activities.

The instructional materials for HMH Into Math, Grade 4 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

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 Spark Your Learning, and Independent Practice, Check for Understanding, 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 3.5, On Your Own, Problem 4, students solve “Aditi takes 72 photos of animals. That is 9 times as many photos as Shane takes. How many fewer photos does Shane take than Aditi? Write equations to model and solve the problem. Let s = the number of photos Shane takes. Let f = how many fewer photos Shane takes than Aditi”. (4.OA.3)
• Lesson 5.7, More Practice/Homework, Problem 6, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. “The Brown family is driving to Junction City, which is 426 miles away. The family drives 60 miles for each of the first 3 hours. Then they drive 55 miles for each of the next 4 hours.  How far are they from Junction City after driving 7 hours?” (4.OA.3)
• Lesson 14.3, On Your Own, Problem 4, “Model with Mathematics, Weston walks $$\frac{1}{4}$$ mile to school and $$\frac{1}{4}$$ mile home. How many miles does Weston walk? Use a visual fraction model, write an equation and find the distance, d”. (4.NF.B.3d)
• Lesson 5.6, Check Understanding, students multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers using strategies based on place value and the properties of operations. “A short book only has 236 words. The library has 5 copies of this book. How many words appear in the books?” (4.NBT.5 and 4.NBT.6)
• Lesson 7.2, On Your Own, Problem 3, students multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers using strategies based on place value and the properties of operations. “Jackie places 552 photos of cats on 4 bulletin boards at the animal shelter. Each board has the same number of photos. How many photos are on each board?” (4.NBT.5 and 4.NBT.6)
• Lesson 7.4, Check for Understanding, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. “There are 48 people going on a hike. Each pack of water has 8 bottles. How many packs are needed for each hiker to have 2 bottles? How can you check that your answer is reasonable?” (4.OA.3)
• Lesson 8.7, More Practice/Homework, Problem 3, students solve multistep word problems posed with whole numbers and having whole-number answers using the four operations including problems in which remainders must be interpreted. “A crane operator moves 6 shipping containers that weigh 215 tons each onto a barge. The same crane operator loads 4 more shipping containers that weigh 194 tons each onto the barge. How many tons of shipping containers did the crane operator load onto the barge? Write an equation to model the situation. How can you check if your answer is reasonable?” (4.OA.3)
• Lesson 14.5, On Your Own, Problem 6, students solve word problems involving addition and subtraction of fractions with the same whole and having like denominators. For example, “Oliver has a board that is $$\frac{10}{12}$$ foot long. After he cuts some off, he has $$\frac{7}{12}$$ foot left. How much did Oliver cut off? Model the problem with an equation and then answer the problem. Use b for the length of board Oliver cut off.“ (4.NF.3d)
• Lesson 15.1, More Practice/Homework, Problem 1, students solve word problems involving addition and subtraction of fractions with the same whole and having like denominators.“Olivia hikes the Appalachian Trail for $$\frac{9}{10}$$ mile on Saturday. She hikes for $$\frac{3}{10}$$ mile on Sunday. How much farther does Olivia hike on Saturday than on Sunday?” (4.NF.3d)
• Lesson 16.2, More Practice/Homework, Problem 1, students solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. “Adam is restoring old wagon wheels and needs to cut 3 wooden spokes that are each $$\frac{5}{8}$$ yard long. What is the total length of wood that he needs to cut? Write an equation using unit fractions that models the problem and the solution.”  (4.NF.4c)
• Lesson 16.3, On Your Own, Problem 9: Model with Mathematics, students solve word problems involving multiplication of a fraction by a whole number. “Lana bakes banana bread for a fundraiser. She uses $$\frac{3}{4}$$ cup of bananas in each loaf. She bakes 5 loaves. How many cups of bananas does she use? Describe a fraction model you could draw to represent the problem. Then model it with an equation and solve the problem.”  (4.NF.4c) 

Each Unit has a Performance Task that involves real world applications of the mathematics from that unit. For example, the Unit 3 Performance Task is about “Visiting New York City”. It has students calculate how much it will cost 20 people to go on a tour of Chinatown (4.OA.1) and for 23 people to go to a show (4.OA.1). It continues by asking students to calculate how much a group comprised of adults and children would save by visiting one attraction versus another (4.OA.2 and 4.OA.3) and to calculate the area of a postcard display. (4.MD.3)

The instructional materials for HMH Into Math, Grade 4 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. The module planning page include a progression 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 5.2, Build Understanding, Problem 1, Part B, builds conceptual understanding of multiplication through the use of area models and the distributive property.  “How can you use the Distributive Property to break apart the base-ten blocks and find the product?” (4.NBT.5)
• Lesson 12.6, On Your Own, Problem 12, emphasizes the application of multi-step problem solving with money. “Four friends earn a total of $7.00 by turning in cans for recycling. If the friends share the amount equally, how much does each get? Give your answer as a decimal dollar amount.” (4.MD.2)
• Lesson 8.6, On Your Own, Problem 6, builds procedural fluency in multi digit multiplication. “Estimate. Then choose a method to find the product. 43 x 35”. (4.NBT.2.

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

• Lesson 14.5, Step it Out, Problem 2, engages students in the application of addition and subtraction of fractions. “Ross makes popcorn at a carnival. At the start of his shift, the container of kernels weighs $$\frac{11}{12}$$ pound. During his shift, he uses $$\frac{3}{12}$$ pound of the kernels. How many pounds of kernels are left after Ross’s shift?” (4.NF.3d)
• Lesson 14.3, Spark Your Learning, students use conceptual understanding to solve application problems. “Caleb enters a frog in a frog-jumping contest. His frog jumps twice. Caleb wants to find the total distance his frog jumps. Explain how can you determine the lengths of each of the frog’s two jumps, then find the total distance the frog jumped.” (There is a visual of a frog jumping, a number line split up into 4 equal parts, with 0 and 1 labeled).
• Unit 5 Performance Task, Problem 1, students solve application problems involving multiplication of fractions by whole numbers. “Enrique lives with his grandmother in an apartment building for senior citizens. He earns extra money by running errands for some of his grandmother’s neighbors. Enrique charges $4 for every $$\frac{1}{4}$$ hour he spends working. He spent $$\frac{2}{4}$$ hour going to the deli for Mr. McGuire, $$1\frac{1}{2}$$ hours delivering papers for the apartment manager and $$\frac{3}{4}$$ hour picking up Mrs. Shultz’s groceries. Did Enrique earn enough money to buy a $49 video game? Explain your reasoning.”
• Lesson 16.3, On Your Own, Problem 9 states “Lana bakes banana bread for a fundraiser. She uses $$\frac{3}{4}$$ cup of bananas in each loaf. She bakes 5 loves. How many cups of bananas does she use? Describe a fraction model you could draw to represent the problem. Then, model it with an equation and solve the problem.” (4.NF.4c)

The instructional materials reviewed for HMH Into Math Grade 4 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, information in the Planning and Pacing Guide also include references to the MPs. For example:

• Correlations in the Planning and Pacing Guide provide lesson examples and to correct over identification. These updates are reflected in the Teacher Edition, both on the introductory pages for each lesson and on the lesson pages with Build Understanding and Step It Out tasks.
• The Planning and Pacing Guide outlines for teachers where to look for each of the SMPs. It states that: “MP1, MP2, 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 in Lesson 16.4; MP2 in Lesson 16.4 and Lesson 14.5; MP3 in Lesson 6.2 and Lesson 12.6; MP4 Lesson 3.3 and Lesson 7.4; MP5 in Lesson 14.5 and Lesson 13.6; MP6 in Lesson 9.2 and Lesson 7.3; MP7 in Lesson 4.5 and Lesson 3.3; and MP8 in Lesson 9.2 and Lesson 11.1.
• 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 about 15 locations listed of where the teachers can look for specific MPs. Examples are given by reference numbers to specific modules and lessons. Each math practice is also categorized as “In every lesson, In every Spark Your Learning and in most lessons, or in most lessons.”
• 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.

Within the Teacher Edition, in the margin under Homework & Test Prep, there is a section that describes the MPs that can be found within the Homework worksheet for the students. For example: 

• Lesson 16.3, identifies MP2 and MP7 in the Lesson Focus and Coherence section, and MP7 is identified in Build Understanding.
• Lesson 12.5, identifies MP6 and MP8 in Lesson Focus and Coherence. MP8 is identified in the Step It Out Section.  
• In Module 15, the Teacher Edition indicates that Lessons 15.1 through 15.4 promote MP1. 

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

• MP.1 is labeled as in every lesson, but there are no explicit connections in Lessons 1.5, 4.1, 5.2.
• Lesson 20.1 indicates that MP3 will be addressed. However, the lesson references MP4.
• In the Planning & Pacing Guide it states that MP2 is in every Spark Your Learning section of the series. However, in the Teacher Edition this is not identified on the Spark Your Learning Page. There is a missed opportunity to make a connection between the Spark Your Learning and MP2 as identified in the Planning and Pacing Guide. The Teacher Edition for the Spark Your Learning provides an Advancing or Assessing question to have students choose strategies and tools. Additionally, Build Shared Understanding recommends to teachers to have students share strategies and tools used to solve the problem.
• Multiple problems within a lesson include MP language with no direct connection to MPs. For example, Lesson 4.7, Problem 11, On Your Own  states “Use structure” language directly from MP7. These MP phrases are in all modules and are in bold prior to the question or problem being posed. For example, Lesson 12.1, Problem 2, On Your Own, states “Critique Reasoning” prior to the problem posed.

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

• Lesson 7.4, Build Understanding, identifies MP2 as students use base-ten blocks to show division.  
• Lesson 12.5, Step It Out, MP8 is identified and connected to students describing connections between coin values and ones, tenths, hundredths.
• Lesson 16.3, Build Understanding, identifies MP4 as students multiply fractions. 

The instructional materials reviewed for HMH Into Math Grade 4 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 13.6, Activate Prior Knowledge, Problem of the Day, “How can you show 2 wholes as a fraction? Draw a visual model to support your answer.” 
• Lesson 6.1, On Your Own, Problem 4, “Write a division equation to model the problem.” 
• Lesson 8.3, On Your Own, Problem 2, “Write and solve an equation for the area model.” 

Examples of MP.5 being identified, where students do not choose tools strategically, as the tools are given to students include: 

• Lesson 12.1, On Your Own, Problem 4, students place a mixed number on a number line. The number line is provided. 
• Lesson 14.4, Step it Out, Problem 2 states “Ruby has $$\frac{2}{3}$$ yard of string. She only needs $$\frac{1}{3}$$ yard to tie a knot. how much string will Ruby have after she cuts off $$\frac{1}{3}$$ yard?” the margin includes tools (pencil, fraction bar and number line) limiting student’s opportunity to choose tools strategically.
• Lesson 15.5, On Your Own, Problem 8,  students explain how the Commutative and Associative Properties are used to add fractions and mixed numbers mentally in the problem such as “Dylan wants to solve $$\frac{3}{8}+1\frac{7}{8}+2\frac{5}{8}$$. Explain how he can use the Commutative and Associative Properties to add fractions and mixed numbers mentally.”

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

• MP1: In Lesson 1.1, Spark Your Learning, states “Some museums keep collections of insect specimens as a historic record. Experts keep track of inventory, inspect specimen cases for damage, and check that labels are set correctly. How can you show the number of beetles in the Museum Insect Inventory?”  Persevere, the Teacher Edition states, “If students needs support, guide them by asking: “What is the number you want to show? How can you use what you already know to show the number to the left of the hundreds place?”
• MP2: In Lesson 3.2, On Your Own, Problem 2, students reason abstractly and quantitatively to answer, “Cory makes these finger puppets. Jackie makes 6 times as many puppets as Cory. How many puppets does Jackie make? Is this an additive or multiplicative comparison problem? Explain how you know without solving.”
• MP7: In Lesson 5.4, Spark Your Learning, students look for and make use of structure to solve,  “Ramy is in a Swim Club that helps children learn about competitive swimming. In one race, he swims 4 laps of the pool. Each lap is 24 feet long. How far does Ramy swim during the race?” Turn and Talk asks, “How could thinking about place value help you solve this problem?”
• MP8: In Lesson 8.5, Step It Out, students look for repeated reasoning to solve, “A celebration concert of the piano ensemble class included students playing in unison on 12 standard pianos. A standard piano has 88 keys. How many keys needed to remain in sync throughout the performance? Write the problem vertically. Then use place value and regrouping to find the answer.”

The instructional materials reviewed for HMH Into Math Grade 4 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

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 those problems labeled “Critique Reasoning.” 

• Lesson 12.6, On Your Own, Problem 7, “Blake has $5.00. He sees some pencil packets that are $1.05. Blake says that he can buy 5 packets. Is he correct? Explain.” 
• Lesson 7.2, On Your Own, Problem 11, “Mara completes this division. Is her answer correct? Why or why not?” 
• Lesson 6.1, On Your Own, Problem 4, “Max says 20 objects can be separated into 4 equal groups. Mara says 20 objects can be separated into 5 equal groups. Who is correct? Explain. Draw to show your answer.”
• Lesson 11.1, Turn and Talk, “Why is it important that the size of the fuel tanks in the go-karts is the same?” and “The fractions $$\frac{4}{5}$$ and $$\frac{7}{8}$$ each have one piece missing from the whole. How can you use the sizes of the missing pieces to compare the two fractions?”
• Lesson 15.1, Turn and Talk, “One classmate represents this problem with an addition equation and another uses a subtraction equation. Who is correct and how do you know?”
• Lesson 15.4, Turn and Talk, “A classmate says that you just need to subtract the whole number from the whole number and the fraction from the fraction to solve this problem. How would you respond?”

The instructional materials reviewed for HMH Into Math Grade 4 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. Lesson 21.1 “Why is it helpful to change hours to minutes? How could you use reasoning to solve this problem without using the table?” 
• Critique, Correct, and Clarify is a strategy used to assist students in constructing viable arguments. Lesson 18.2, On Your Own, Problem 5, “Point out to students that Problem 5 has an error. Encourage students to describe the error and review explanations with a partner. Students should refine their responses after their discussions with a partner.” In Lesson 19.2, On Your Own, Problem 19, students analyze the reasoning of a fictitious student. Teacher guidance says “Point out that in Problem 19 Jimmy’s reasoning is not complete. Encourage students to describe why his reasoning is incomplete and review explanations with a partner. Students should refine their responses after their discussions with a partner.”
• Lesson 4.1, Connect Math Ideas, Reasoning and Language states “Select students who used various strategies and tools to share with the class how they solved the problem. Have students discuss why they chose a specific strategy or tool.”
• Lesson 1.5, Optimize Output, “Point out to students that the Turn and Talk asks how it is possible for two different estimates to both be correct. Encourage students to describe the thought processes of Anja and Liam and review explanations with a partner. Students should refine their responses after their discussions with a partner.”
• The Teacher Edition includes Turn and Talk in margin notes to prompt student engagement.  Lesson 3.1, s "Encourage students to discuss how their visual models show 3 groups of 4, which is the same as showing 4 three times. If some students are struggling, pair them with the students who quickly found a different visual model.”

The instructional materials reviewed for HMH Into Math Grade 4 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 11.2 “A known size or amount that helps you understand a different size of an amount is a benchmark. Common benchmarks are 0, $$\frac{1}{2}$$, 1, and $$1\frac{1}{2}$$”. 
• 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. Module 8: “By giving all students regular exposure to language routines in context, you will provide opportunities for students to listen, speak, read, and write about mathematical situations and develop both mathematical language and conceptual understanding at the same time”.
• In the lesson planning pages, Sharpen Skills in some lessons include Vocabulary Review activities. Lesson 20.1 “Objective: Students review vocabulary used to classify triangles by angles and by sides. Materials: markers, poster paper.” The lesson continues “Have students work in small groups to create a poster. Have students divide their posters into two sections: Classify Triangles by Sides and Classify Triangles by Angles. In the first section, have students write the terms scalene, isosceles, and equilateral. In the second section, have them write the terms acute, right, and obtuse. Have students draw an example of each type of triangle and write a definition for each term. Have each group share their poster with another group.”
• Guide Student Discussion provides prompts related to understanding vocabulary. Module 1: “Listen for student who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms place value, greater than, less than, equal to, and compare. Ask students what they mean if they use those terms. Rounding to the nearest 10, what numbers round to 860? What whole numbers round to 900 when you round them to the nearest hundred? How can you determine which numbers to place in the table?”
• Vocabulary is highlighted and italicized within each lesson in the materials. 
• There is a vocabulary review provided 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 4 meet the expectations that there is a clear distinction between problems and exercises in the materials.

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

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 16.2, Spark Your Learning students experiment with the following problem. “Karl participates on a youth wheelchair-basketball team. The team practices 3 times a week after school for $$1\frac{3}{4}$$ hour. How can you use multiplication to find how long Karl’s basketball team practices each week?” Students then use fraction tiles or fraction circles to find a solution. In the Build Understanding students develop understanding of multiplication of a whole number and a fraction. Then in the On Your Own section student list the first 4 multiples of $$1\frac{7}{10}$$. 

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

Overall, lessons are intentionally 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 section of Lessons students use manipulatives and/or visual models to experiment the mathematics thus developing a concrete or representational understanding. This is followed by a Turn and Talk with a partner allowing students to process the connections they have found. Throughout the lessons, students are provided scaffolding with new content in the Build Understanding and Step It Out sections, where the abstract concept is broken down into smaller steps with additional Turn and Talk opportunities, and 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 4.1, students begin the Spark Your Learning by using manipulatives or a visual model to solve the following problem: “Tickets to the school play just went on sale. If the fourth-graders buy 100 tickets. How much will they spend? Come and See the School Play Tickets $5 each” (In the Build Understanding they experiment with the relationship between 7 and 70. They continue to practice 3 x 1, 3 x 10, 3 x 100 and reflect on “What is the same about all the equations?” In the On Your Own, they work with the abstract concept of 5 x 4 hundreds =________hundreds.

Concepts are developed through the modules such as 4.NBT.2.6 which is introduced in Module 4 with estimating quotients, then developed in Module 6 and 7 with a study of division by 1 digit numbers.

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

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 9.1, Homework has 6 problems. The first problem asks students to construct an argument given constraints about the size of a piece of fabric and if there is enough information to find the area. The next four problems are fluency problems with finding area and the last problem asks students to compare two sets of garden plans to determine which has the greater area. 

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

Lessons indicate what materials/manipulatives will be needed. Lessons on the student pages incorporate a picture of the manipulative that they will use. For example, in Lesson 4.1, students use two color counters to represent the following problem: “There are 8 bags of pencil-top erasers for the school store. How many erasers are there if each bag has 70 erasers? 700 erasers?” The manipulatives provide opportunities for students to develop a conceptual model of problems that they will then represent in pictorial form in their student workbook. 

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

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

The materials rely 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 4 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 12, Are you Ready, begins with students counting coins and includes a visual image of quarters, nickels and times, needed to find the value.

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

Throughout the Teacher Edition questions are posted to help support teachers with questions to guide students’ mathematical development. 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 8.6, Activate Prior Knowledge, “What is one strategy you could use when you multiply by 7?” In the same lesson the Step It Out Turn and Talk states, “How can you use the Distributive Property to solve the problem?” 
• Lesson 20.2, Spark Your Learning, includes questions in the margin notes:  How could you use classroom objects to act out the problem?” “What visual models can you draw to represent lengths or comparison?” and “What comparison words could you use to describe the length?”  
• Lesson 13.4, Build Understanding, provides two questions: “How could you figure out what angle measure is equivalent to one-fourth of a circle?” and “If you divide by 360 degrees by 4 to determine the angle measurement of $$\frac{1}{4}$$ of a circle, how can you determine what $$\frac{1}{9}$$ of a circle measures?”

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

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 11.3, Plan for Differentiated Instruction section includes three columns titled, On Track, Almost There, and Ready for More.”The guidance for teachers is designed to be used in small group instruction.
• Lesson 14.4, Activate Prior Knowledge, includes 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 3, Lesson 15.2 or Complete the Prerequisite Skills Activity.” 

The instructional materials for Into Math Grade 4 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. 

This materials 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 11.3 this table lists the 3rd grade standard that supports the 4th grade on level standard and what 5th grade standard this will lead into. The explanation is a brief set of bullets and does not include any tasks or examples for the teacher. 
• Professional Learning notes are present in each lesson. In Lesson 11.3, Professional Learning, discusses “Using Mathematical Practices and Processes.” 
• The “Planning and Pacing” book provided in the series includes a correlation chart for the math practices that defines each math practice in full.

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

Each module in the Teacher Edition includes Mathematical Progressions 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 4 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 4 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 4 includes a Math on the Spot video for Lesson 2.

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

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

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

• The module overview in the Teacher Edition contains “Common Errors” as students engage in an introductory task and provides questioning strategies intended to build student understanding.
• The Spark Your Learning planning page for each lesson in the Teacher Edition includes a Common Error section related to the content of the lesson 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 4 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 4 meet the expectations that assessments clearly denote which standards are being emphasized.

The Lesson Focus and Coherence page indicates the CCSSM that will be addressed within the Lesson. Throughout the lesson 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 4 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 4 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 saying “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 4.1 the Learning Mindset asks “What do I already know that can help me multiply by tens, hundreds, and thousands on our own?”

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

• At the beginning of each module, Teaching for Depth provides information on strategies to use when teaching the concept, including Represent and Explain, which focuses on ways for students to describe and picture a concept, or Make Connections, which helps students understand a new idea by connecting it to previous knowledge.
• At the beginning of each module, Mathematical Progression 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 4 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

• There are Reteach and Challenge activities for each lesson.
• Each module includes Plan for Differentiated Instruction that provides teachers with teacher-guided, Small-Group Options and self-directed Math Center Options based on student need: On Track, 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 4 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 5.4, Spark Your Learning has this prompt: “The Monstrosity Roller Coaster has seats for 136 riders. The roller coaster completes 4 runs each half hour.  If all the seats on the roller coaster are filled each run, how many people can ride in a half hour.” This problem provides multiple entry points and solution strategies for students. However, Spark Rour Learning is not present in every lesson. 

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

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

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

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

• Language Objectives are included in every lesson.
• 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 4 meet the expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

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

• 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 4 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 4 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 4 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 4 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 4 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 4 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 4 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 4 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 4 do not incorporate technology that provides opportunities for multiple students to collaborate with the teacher or one another.