## HMH Into Math

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### Overall Summary

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for alignment to the Common Core State Standards (CCSS). ​The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor. The materials partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Cluster Standards for Mathematical Practice (MPs).

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

### Focus & Coherence

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

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

​The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for not assessing topics before the grade level in which the topic should be introduced. The materials assess grade-level content and, if applicable, content from earlier grades.

##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for HMH Into Math Grade 8 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 Interim Growth assessments.

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

• Module 1, Form A, Question 7 states: “Graph the image of the triangle after it is translated 4 units down and reflected across the y-axis.” (8.G.3)
• Unit 6 Performance Task states: “Kevin makes a cylinder-shaped candle. He uses a metal can as a mold. The can has a radius of 6 centimeters and a height of 18 centimeters. How much wax is needed to fill the metal can? Show your work.” (8.G.9)
• Module 7, Form B, Question 8 states: "Brianna is comparing the price for two different bicycle rental companies. Option A charges a fixed fee and an hourly rate. Option B charges per hour with no fixed fee. The system of equations shows the total cost, y, for x hours. [system given: y = 6x + 16; y = 14x] Part A: Graph the system of equations. Part B: What is the cost in dollars  when the two plans cost the same amount?" (8.EE.8c)
• Module 11, Form A, Question 8 states: “Two boards are resting against each other to form an isosceles triangle as shown in the diagram. What is the height, h, rounded to the nearest hundredth in.?” (8.G.7)

#### Criterion 1.2: Coherence

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade.

##### Indicator {{'1b' | indicatorName}}
Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for spending a majority of instructional time on major work of grade.

• The number of modules devoted to major work of the grade is 9 out of 13, which is approximately 69%.
• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 39 out of 50, which is approximately 78%.
• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 101 out of 132 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 78% of the instructional materials focus on major work of the grade.

#### Criterion 1.3: Coherence

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The instructional materials are also consistent with the progressions in the standards and foster coherence through connections at a single grade.

##### Indicator {{'1c' | indicatorName}}
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for HMH Into Math Grade 8 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:

• In Lesson 8.3, students use scatter plots and equations of trend lines to analyze data. For example, On Your Own Question 8 states: "According to the equation of the trend line, each additional hour spent studying raises test scores by how much? How do you know?" (8.SP.3 and 8.EE.5)
• In Lesson 11.3, students use the Pythagorean Theorem (8.G.7) to find side lengths and rational approximations of irrational numbers (8.NS.2) to determine appropriate solutions. For example, given a 2m x 2m x 2m cube, “What is the longest rod that can fit in this cube? Round to the nearest tenth and show your work.”
• In Lesson 11.3, students apply the Pythagorean Theorem (8.G.2) to calculate the height of a cone (8.G.9). For example, given the radius of Cone A is 10 cm and the slanted side is 17 cm and the radius of Cone B is 12 cm and the slanted side is 18 cm, “Which is taller, Cone A or Cone B” By how much? Round to the nearest hundredth.”
• In Module 13, students solve problems about the volumes of cones, cylinders, and spheres (8.G.9) using square and cube roots (8.EE.2).
• In Lesson 13.1, students find the volume of a cylinder (8.G.9) with dimensions expressed in scientific notation (8.EE.4) and express the volume in scientific notation.
• In Lesson 13.2, students use the formula for the volume of cylinders (8.G.9) to model relationships between quantities (8.EE.5 and 8.F.4): “Consider a set of cones that all have a radius of 1 centimeter. The heights of the cones are 1 centimeter, 2 centimeters, 3 centimeters, 4 centimeters, 5 centimeters. A) Complete the table. Leave the volumes in terms of pi. B) Is the relationship in the table a proportional relationship? C) Write an equation that gives the volume y of a cone with radius 1 centimeter if you know the height x of the cone. Describe the graph of the equation.”
##### Indicator {{'1d' | indicatorName}}
The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials for HMH Into Math Grade 8 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 with assessments can be completed in 154 days: 106 days for lessons and 48 days for assessments.

• The Planning and Pacing Guide and the Planning pages at the beginning of each module in the Teacher Edition provide the same pacing information.
• Grade 8 has six units with 13 modules which contain a total of 50 lessons.
• The pacing guide designates all 50 lessons as 2-day lessons, leading to a total of 100 lesson days; a “day” is defined as one period in a traditional structure.
• Each unit includes a Unit Opener which would take less than one day. There are six Openers for Grade 8 (6 days).
• This is a total of 106 lesson days.

Assessments included:

• The Planning and Pacing Guide (page 20) indicates a Beginning, Middle, and End of Year Interim Growth test that would require 1 day each (3 days).
• Each module starts with a review assessment titled “Are You Ready?” There are 13 modules (13 days).
• Each unit includes a Performance Task which indicates an expected time frame ranging from 25-45 minutes. There are six Performance Tasks for Grade 8 (6 days).
• Each module has both a review and an assessment. There are 13 modules (26 days).
• Based on this, 48 assessment days can be added.
##### Indicator {{'1e' | indicatorName}}
Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials for Into Math Grade 8 meet expectations for being consistent with the progressions in the Standards. In general, the materials identify content from prior and future grade-levels and relate grade-level concepts explicitly to prior knowledge from earlier grades. In addition, the instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems.

• In the Teacher Edition, the introduction for each module includes Mathematical Progressions, which lists standards under the areas of Prior Learning, Current Development, and Future Connections and clarifies student learning statements in these categories. For example, Module 7 System of Linear Equations states, “Prior Learning: Students wrote and solved one-variable equations.” (7.EE.4); “Current Development: Students use graphing to determine the approximate solution to a system.” (8.EE.8); “Future Connections: Students will represent constraints by systems of equations.” (A-CED.3)
• In Activate Prior Knowledge at the beginning of each lesson, content is explicitly related to prior knowledge to help students scaffold new concepts.
• In Lesson 9.2, Spark Your Learning states, “Using what you know about proportion and percent, what information can you determine from the table?”
• Each module includes a diagnostic assessment, Are You Ready?, that explicitly identifies prior knowledge needed for the current module. For example, in Module 3, Are You Ready? addresses solving one-step equations (6.EE.7) and writing and solving two-step equations (7.EE.4a).
• In the Module Opener, "Assess Prerequisite Concepts" identifies content as prerequisite knowledge including the Grade and Module where the prerequisite concept is addressed.

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

• In the Planning and Pacing Guide, there is a Correlations chart that outlines the mathematics in the materials. According to this chart, all grade-level standards are represented across the 13 modules.

Within each lesson, Check Understanding, On Your Own, and More Practice/Homework sections include grade-level practice for all students. Margin notes in the Teacher Edition also relate each On Your Own practice problem to grade-level content. Examples include:

• In Lesson 5.4, Question 9 states, “Given a graph and a graphic with sizes labeled, students solve “A rain barrel and a cistern are filling at constant rates. The amount of water in the rain barrel over time is shown in the graph. The amount of water in the cistern is given by y = 200x, where x is the time in hours and y is gallons of water. A) How fast is the rain barrel filling? How fast is the cistern filling? B) If both the barrel and the cistern were empty at t = 0, which would completely fill first? Explain.” (8.EE.5)
• In Lesson 11.1, More Practice Question 5 states, ‘As the students explain a proof of the Pythagorean Theorem, they solve “A) Find the missing side of a triangle with leg length 5 yards and hypotenuse 13 yards. B) Find the missing side of a triangle with leg length 30 yards and hypotenuse 78 yards. C) How are the lengths of the leg and hypotenuse in Part A related to the lengths of the leg and hypotenuse in Part B? D) How is the length of the missing side in Part A related to the length of the missing side in Part B?” (8.G.6)

When work is differentiated, the materials continue to develop grade-level concepts. An example of this is Lesson 4.3, which involves work with transversals and identification of angles. The corresponding Reteach page provides guided notes for students to follow in order to access the concept and the Challenge page has two parallel lines cut by two transversals where students must solve for x using algebraic expressions given in three angles.

##### Indicator {{'1f' | indicatorName}}
Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for HMH Into Math Grade 8 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:

• In Lesson 7.2, the learning objective is “Solve a system of two linear equations by graphing,” shaped by 8.EE.C.
• In Lesson 8.2, the learning objective is “Use trend lines to describe a linear relationship between two variables,” shaped by 8.SP.A.
• In Lesson 12.1, the learning objective is “Develop and use the properties of integer exponents,” shaped by 8.EE.A.
• In Lesson 5.3, both Learning and Language objectives are “Graph proportional relationships; Interpret unit rate as the slope of the graph of a proportional relationship” and “Explain how to find the unit rate of a proportional relationship from graphs and tables and how to determine whether a graph should be continuous or discrete based on the situation it represents,” shaped by 8.EE.B.

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 include:

• In Lesson 4.1, students write and solve an equation (8.EE.C) to determine the measures of the angles of a triangle (8.G.A).
• In Lesson 5.1, students determine that triangles are similar (8.G.A) by finding two pairs of congruent angles and comparing the slopes of the triangles (8.EE.B).
• In Lesson 6.2, students identify linear and nonlinear equations (8.F.A) by examining the slopes and y-intercepts of the equations (8.EE.B).
• In Lessons 11.1 and 11.2, students solve equations for missing side lengths resulting from the Pythagorean Theorem (8.EE.A and 8.G.A).

### Rigor & Mathematical Practices

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for Gateway 2, rigor and balance and practice-content connections. The instructional materials meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and they meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor

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

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The instructional materials also do not always treat the aspects of rigor separately or together.

##### Indicator {{'2a' | indicatorName}}
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials for HMH Into Math Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

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

• In Lesson 6.1, students determine if tables show a functional relationship between two variables and provide an explanation. (8.F.1)
• In Lesson 6.5, students conclude that a table, graph, and description represent the same information, interpret the slope and y-intercept, and use the information to analyze the situation. (8.F.4)
• In Lesson 6.6, students extend their understanding of linear relationships by matching nonlinear graphs with various contexts, sketching nonlinear graphs based on context, and explaining contexts based on provided graphs. In On Your Own, Question 3, students answer, “The graph represents the speed of a swimmer during a race. Describe the swimmer’s speed during the course of the race.” (8.F.5)
• In Lesson 1.5, Build Understanding, Question 1, students use fabric shapes to experiment with transformations by moving the shapes on top of another given figure and “explain how the figures are the same and how they are different.” (8.G.1)
• In Lesson 2.2, students manipulate two-dimensional figures on a coordinate plane to understand dilations. In Step It Out, Question 2, teachers use discussion questions to develop understanding such as, “How does a scale factor greater than 1 affect the dilation? Between 0 and 1? Equal to 1?” (8.G.3)
• In Lesson 5.1, Build Understanding, students develop the concept of slope by examining two similar right triangles with hypotenuses that are on a line including calculating the rate of change. In Step It Out, students continue to investigate the similar triangles. Part E states, “Find the rise-over-run relationship modeled by the line, using the coordinates of Point R and the origin.” Part F relates that ratio to slope. In Lesson 5.4, students compare proportional relationships by calculating and comparing rates to determine the fastest runner. (8.EE.2)
##### Indicator {{'2b' | indicatorName}}
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials for HMH Into Math Grade 8 meet expectations for attending to those standards that set an expectation of procedural skills.

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

• In Lesson 3.1, students solve multi-step equations such as "2(11t+1.5t) = 12-5t." Equations include all operations and various forms of rational numbers, including fractions and decimals. (8.EE.7)
• In Lesson 3.2, students solve linear equations with no solution or infinitely many solutions. They complete partial equations that would have infinitely many or no solutions such as More Practice/Homework, Question 15, which states, “Complete the equation so that it has no solution: 10.5x − 4 = 5 + _______” (8.EE.7)
• In Module 7, students solve systems of linear equations in multiple ways including graphing, elimination, and substitution. For example, in Lesson 7.6, Wrap Up, Question 11 states, “Bowling costs $6 per game and virtual golf costs$0.50 per hole. Bowling takes 30 minutes per game and virtual golf takes 7.5 minutes per hole. Toni spends 1 hours and $8 between the two activities. Write a system of equations relating b, the number of games of bowling played, to g, the number of holes of virtual golf played. Solve the system you wrote and interpret your solution.” (8.EE.8b) • In Lesson 10.1, students convert decimal expansions that repeat into rational numbers. For example, in More Practice/Homework, Questions 6-11 state, given a repeating decimal “Write the number as a fraction or mixed number in simplest form.” (8.NS.1) ##### Indicator {{'2c' | indicatorName}} Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade The instructional materials for HMH Into Math Grade 8 meet expectations for teachers and students spending sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and concepts of the grade-level, and students independently demonstrate the use of mathematics flexibly in a variety of contexts. During Independent Practice and On Your Own, students often engage with problems that include real-world contexts and present opportunities for application. More Practice and Homework contains additional application problems. • In Lesson 6.5, On Your Own, students compare properties of two functions where one is represented by a graph and the other by a table. For example, “Leon and Jacey work for a florist. They each program a drone to deliver boxes of flowers to neighbors. With Leon’s program, he must manually attach the box to the drone. With Jacey’s program, the drone can pick up one box at a time from a pile. How much time does it take Leon to attach a box? How much time does Jacey’s drone take to pick up a box? Which program has the drone traveling faster? How do you know?” (8.F.2) • In Lesson 7.6, More Practice/Homework, Question 3, students find the time needed for one horse to catch a second horse as each moves at a different rate. In Check Understanding, Question 1 states, “The Spartan basketball team scored 108 points in last night’s game. They scored 48 baskets in all, making a combination of two-point and three-point baskets. There were not points due to free throws. How many three-point baskets did the Spartans make?” (8.EE.8c) • In Lesson 11.3, students use the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. For example, in Check Understanding, Question 1 states “Computer monitors are measured diagonally, from corner to corner. If the rectangular screen of a 40-inch monitor is 35 inches wide, what is the height of the monitor? Round to the nearest tenth.” (8.G.7) • In Lesson 3.3, students create and apply open-ended linear equations to real-world scenarios, such as “A business breaks even when their production costs are equal to their revenue. The expression 120 + 4x represents the cost of producing x items. Decide on a selling price for each item and write an expression for the revenue generated by selling all x items. How many items would you need to sell at your chosen price to break even? Write an equation and solve it.” Answers can vary. (8.EE.7) • In Lesson 9.1, students conduct a survey of students in their class and interpret the results of the data by way of a two-way bivariate table. For example, “Conduct a survey of students in your class. Each student that you survey should be asked whether or not the student has a curfew on school nights and whether or not the student has assigned chores at home. Record your survey data in the table. Is there evidence of an association between having a curfew on school nights and having chores? Explain.” (8.SP.4) ##### Indicator {{'2d' | indicatorName}} Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade. The instructional materials for HMH Into Math Grade 8 meet expectations for the three aspects of rigor not always being treated together and not always being treated separately. In general, two, or all three, of the aspects are interwoven throughout each module. The Module planning pages include a diagram showing the first few lessons addressing understanding and connecting concepts and skills and the last lessons addressing applications and practice. All three aspects of rigor are present independently throughout the program materials. Examples include: • In Lesson 2.1, students develop conceptual understanding of dilations. During Build Understanding, students measure the angles and side lengths of a triangle, then draw another triangle with side lengths half of the original. In the remainder of the lesson, students analyze the characteristics of pairs of figures, identify if the pairs are proportional, and identify if the pair represents a reduction or enlargement. • In Lesson 11.3, On Your Own Question 4, students apply the Pythagorean Theorem to find the length of the longest diagonal in a wardrobe closet that measures 36 inches by 24 inches by 96 inches. • In Lesson 12.1, students develop procedural skill in using properties of exponents. In Build Understanding, students write expanded form of exponents and use prior knowledge to simplify expressions. Within On Your Own and More Practice/Homework problems, students apply these properties to simplify expressions such as $$\frac{9^{-2}\centerdot 9^7}{9^3}$$. Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include: • In Lessons 8.1-8.3, students develop procedural skill in creating scatter plots by graphing relationships between two variables and using the plot to draw and analyze trend lines. Students apply trend lines to interpreting data in real-world contexts. For example, “The average length of an animal’s life is related to the average length of pregnancy for that type of animal. The scatter plot shows data for several different types of animals. A) Sketch a trend line to model the data. Circle the outliers and explain how they influence your choice for the trend line. B) How would the trend line change if the outliers were removed? C) The average length of pregnancy for a polar bear is 240 days. Use the trend line to estimate the average length of a polar bear’s life.” • In Lesson 13.1, students build conceptual understanding in Spark Your Learning by comparing the volume of a rectangular prism to the volume of a cylinder in order to derive the formula for a cylinder. Students develop procedural skill by applying the formula to solve multiple problems involving volume of a cylinder. For example, “The height of the cylindrical container shown is 7 inches. Find the volume of the cylinder.” Additional questions provide diagrams of cylinders for students to find the volume. #### Criterion 2.2: Math Practices Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified and clearly labeled throughout the materials, and the instructional materials support the standards’ emphasis on mathematical reasoning. ##### Indicator {{'2e' | indicatorName}} The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade. The instructional materials reviewed for HMH Into Math Grade 8 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade-level. All eight MPs are clearly identified throughout the materials, including: • MPs are identified in both the Planning and Pacing Guide and the Teacher Edition. • The Planning and Pacing Guide explains each MP and provides a correlation to specific lessons. • MP1 is correlated with “every lesson,” but it is not identified in the Focus and Coherence pages of the Teacher Edition for each lesson with other identified MPs. • The Teacher Edition labels an MP for the Build Understanding and Step It Out tasks. • The Module Review includes a question labeled "Use Tools" in the student edition where students choose a tool and explain their choice. Examples of the MPs being used to enrich the mathematical content include: • Lesson 10.1 identifies MP2 and MP7 as the focus MPs for the lesson. The materials identify MP2 with Build Understanding and MP7 with Step It Out. • Some lessons include an explanation about the connection to the MP in Professional Learning. For example, in Lesson 6.3, the explanation for MP6 states, “This lesson calls for students to analyze numbers associated with problems arising in everyday life, society, and the workplace. Students use graphs, equations and tables to correctly assign meaning to numbers, by also attending to the precise wording of a scenario. Then students interpret the meaning of these numbers through calculation of slope and y-intercept. Finally, students use this information to draw conclusions and compare options. In this way, students are able to make informed choices based on correct mathematics.” ##### Indicator {{'2f' | indicatorName}} Materials carefully attend to the full meaning of each practice standard The instructional materials reviewed HMH Into Math Grade 8 partially meet expectations for carefully attending to the full meaning of each practice standard (MP). The materials do not attend to the full meaning of MP4 and MP5. For MP4, mathematical models are provided for students, and they use tools as directed by the materials, examples include: • MP4: In the lesson planning pages for Lesson 6.4, MP4 is identified as the focus of the lesson including the description of the MP, which states “Can you show me how you solved the problem using your model? What is another model you could use to solve the problem?” Students are directed to use the y-intercept equation to model the situations given, including being provided with fill-in-the-blank processes to find the slope. • MP5: In Lesson 1.4, students use tracing paper, a ruler, and a protractor (all given) to investigate rotations preserving length and angle measure. • In the Spark Your Learning problems and most module reviews, there are general notes for teachers to ask the students. For example, “If students need support, guide them by asking, Which tool could you use to solve the problem? Why is this tool more strategic?” or “State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.” Examples of the instructional materials attending to the full meaning of the MPs include: • MP1: In Lesson 2.2, Spark Your Learning states, “On a computer, Raquel uses polygons and a circle to make a model of the top of London’s Big Ben clock tower. Then she reduces it. Compare Raquel’s image and reduction. What did you find?” In Persevere, the Teacher Edition states, “If students need additional support, guide them by asking: Are parallel lines in the preimage parallel in the image? perpendicular lines? What tool can you use to measure the angles of the top triangle? Why do angle measures remain the same when reducing the image? Advancing Use Tools: Which tool could you use to solve the problem? Why is this tool more strategic?” • MP2: In Lesson 7.6, Question 2, students reason abstractly and quantitatively to determine the cost of two types of plants. The question states, “Mr. Chen buys 5 tomato plants and 3 cucumber plants for$33.00. His neighbor buys 4 tomato plants and 2 cucumber plants of the same varieties for \$24.90 at the same nursery. What is the cost of each type of plant?”
• MP6: In Lesson 5.1, Question 5 states, “Attend to Precision. Line L passes through the origin and the point (4, 5). Suppose point (x, y) also lies on Line L. A) The slope of Line L from the origin to (4, 5) is ____. B) Why is the slope of Line L from the origin to (x, y) the same as the slope from the origin to (4, 5)?”
• MP7: In Lesson 3.1, Step It Out, students solve an equation with decimal coefficients and use the structure of the coefficients to determine a different way to solve the equation. Part A states, “Look at the decimals in the equation and think about how you could rewrite the equation with integer coefficients. What is the least power of 10 you could multiply each term by to eliminate all the decimals?” Students write an equivalent equation by multiplying the coefficients by a multiple of 10, solve the equivalent equation, and compare the solution to the original problem. Part D asks, “Which equation was easier to solve? Why?”
• MP8: In Lesson 12.1, Spark Your Learning, students are presented with a situation about a rope that is $$2^4$$, or 16 feet long and repeatedly cut in half until each piece is $$2^0$$ or one foot long. The question states, “What would happen if Alex took a 1-foot section of rope and continued this process? What pattern do you notice?”
##### Indicator {{'2g' | indicatorName}}
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
##### Indicator {{'2g.i' | indicatorName}}
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

• In Lesson 4.1, Build Understanding, students investigate of the sums of the angles of triangles. Part B asks, “What do you notice about the sum of the measures of the three triangles?” Part C asks, “Do you think this is true for all triangles? Explain.”
• In Lesson 4.1, Question 5 states, “Can a triangle have two obtuse angles? Explain your answer.”
• In Lesson 7.1, Question 4C states, “If the plan is to use the tablet for eight months, which tablet costs less overall? Explain.” The suggested answer is “Brand B is a better deal since the graph lies below the graph for Brand A whenever x is greater than 4.”
• Lesson 7.5 states, "For Problems 2–4, use the system of equations shown. {3x +5y = 4, 6x + 10y = 4} 2) Does the system have a solution? Explain. 3) Manuel says that he can change the 4 in the second equation to any number and the system will have no solution. Is Manuel correct? Explain. 4) How can one number in the second equation be changed so the system has only one solution? Explain."
• In Lesson 9.3, Question 2 states, “Is there an association between the mouse being male or female and the color of the fur for the sample? Explain.”
• In Lesson 9.3, Question 8 states, “The two-way relative frequency table shows data from a mayoral election. Chris said there is no association between the candidate a person voted for and whether or not they support the ballpark because 55% voted for Chan but 80% support the new ballpark. Do you agree or disagree? Why?”
• In Lesson 13.2, Question 16 states, “The cone and cylinder in the figure below have the same radius. The height of the cylinder is 3 times the height of the cone. Jared looked at the figure and concluded that the volume of the cylinder must be 3 times the volume of the cone. Therefore, he said the volume of the cylinder is 3 ✕ 40, or 120 cubic centimeters. Do you agree with Jared’s reasoning? Explain.”
##### Indicator {{'2g.ii' | indicatorName}}
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for HMH Into Math Grade 8 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.

• The Teacher Edition provides Guided Student Discussion with questions to encourage students to explain their thinking. For example, Lesson 1.3 states, “What happens to the order of the vertices ABCD when the parallelogram is reflected over the x-axis? What about the y-axis?” and Lesson 8.2 states, “Compare the trend lines sketched by several other students. What do you notice about these lines? How would changing the slope or y-intercept of your trend line sketch affect predictions made by the trend line?”
• Turn and Talks are provided multiple times per lesson. For example, in Lesson 1.1, Turn and Talk states, “Describe multiple ways you could move the original triangle to tile the floor.” Teachers are given a possible answer as well as additional guidance to assist students in constructing arguments, for example, “If some students have trouble explaining how they moved the triangular tiles, encourage them to use positional language, such as left, right, above, below. Students should also be able to describe the direction of the tiles.”
• The Teacher Edition includes Communicate and Collaborate in margin notes to prompt student engagement. For example, Lesson 6.5 states, “Select students who used various strategies and have them share with the class how they solved the problem. Ask students to share with each other the reasons they prefer a particular representation. Encourage students to share how the representation is helpful and what information they got from it that helped them solve the problem. Encourage students to ask questions of their classmates.”
• The Teacher Edition also provides Cultivate Conversation prompts in the lessons. For example, Lesson 7.2 states, “Stronger and Clearer Each Time. Have students share how they solved the problem. Remind students to ask each other questions of each other that focus on how they approached the problem. Then have the students refine their answers.”
• In the margin notes for practice questions that are identified as a mathematical practice, there is an explanation about why that practice is labeled. For example, in Lesson 7.5, Question 2-4 are labeled “Critique Reasoning” and the notes explain, “students have the opportunity to critique reasoning and demonstrate an understanding of coefficients in systems of linear equations that have no solution, one solution, or infinitely many solutions.”
• In lesson planning pages, sometimes Professional Learning provides a rationale for a lesson labeled “Using Mathematical Practices and Processes.” For example, Lesson 11.2, which is labeled MP3, states, “This lesson provides an opportunity to address this Mathematical Practice Standard by focusing on proving the converse of the Pythagorean Theorem. Students begin by co-construction an informal proof, or argument, to prove that the converse of the Pythagorean Theorem is true. Students then apply this to various situations to determine if a triangle is a right triangle based on the measurement of each side, in and out of context. Also the opportunity is given to critique the arguments and reasoning of others about whether or not a triangle can be classified as a right triangle.”
• Lesson 10.2 states, “Critique, Correct, and Clarify. Have students share their ideas about the side length of a cube with a volume of 50 $$cm^3$$. Encourage students to describe a possible side length for this cube and review explanations with a partner.”
##### Indicator {{'2g.iii' | indicatorName}}
Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for HMH Into Math Grade 8 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.

• Key Academic Vocabulary is listed at the beginning of the module in a table that includes any prior vocabulary relevant to the lesson and new vocabulary.
• Each lesson includes a Language Objective that emphasizes mathematical terminology. For example, Lesson 4.1 states, “Describe angle relationships in triangles.”
• In Module planning pages, there is a Linguistic Note on the Language Development page which provides teachers with possible misconceptions relating to academic language. For example, Module 1 states, “Listen for students who do not distinguish the difference between the two terms transformation and translation, and the two terms reflection and rotation. These pairs of words are very similar and their pronunciation can be easily confused. Have these students model a translation, a reflection, and a rotation and have them say the word to describe each transformation. Make sure students can correctly use the word transformation to describe any translation, reflection, or rotation of a figure. Model the correct language for students.”
• In Sharpen Skills in the lesson planning pages, some lessons include Vocabulary Review activities. For example, in Lesson 3.3, students have two activities: “1) Draw a shape on graph paper and dilate the image. Use the drawings to define the terms: image, preimage, enlargement, reduction, dilation, center of dilation, similar. 2) Draw a Venn diagram labeled Dilation and Rotation. Fill in the Venn diagram with the terms: image, preimage, similar, congruent, reduction and enlargement.”
• Guided Student Discussion often provides prompts related to understanding vocabulary, such as “Listen for students who correctly use review vocabulary as part of their discourse. Students should be familiar with the terms polygon, vertex, ordered pair, x-coordinate, y-coordinate, and integer. Ask students to explain what they mean if they use those terms.”
• Student pages include vocabulary boxes that define content vocabulary.
• Vocabulary is highlighted and bold within each lesson in the materials. Terms are highlighted in blue if it is meant to review and yellow if it is new vocabulary.
• There is a vocabulary review at the end of each module where students match new vocabulary terms with their meaning and/or examples provided, fill-in-the-blank with definitions or examples, or create a graphic organizer to help make sense of terms.
• The Teacher Edition sometimes suggests creating an Anchor Chart to “connect math ideas, reasoning, and language” where students define terms with words and pictures, trying to make connections among concepts. For example, Lesson 4.1 has a sample anchor chart that includes vocabulary related to angle relationships.
• There is an Interactive Glossary at the end of the text where the definition and a visual (e.g., diagrams, symbols, etc.) are provided for each vocabulary word. In the student book, the instructions read, “As you learn about each new term, add notes, drawings, or sentences in the space next to the definition. Doing so will help you remember what each term means.”

### Usability

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Use & Design

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

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for being well-designed and taking into account effective lesson structure and pacing. The instructional materials include an underlying design that distinguishes between problems and exercises, assignments that are not haphazard with exercises given in intentional sequences, variety in what students are asked to produce, and manipulatives that are faithful representations of the mathematical objects they represent.

##### Indicator {{'3a' | indicatorName}}
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials for HMH Into Math Grade 8 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.

Additional practice is provided in a separate student journal which allow students to independently show their understanding of the material. Exercises and problems are designed for students to demonstrate understanding and skills in application and non-application settings. Test Prep and Spiral Review also include exercises.

##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials for HMH Into Math Grade 8 meet the expectations that the design of assignments is intentional and not haphazard.

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

• Spark Your Learning serves to motivate and set the stage for students to learn new material and persevere through a related mathematical task.
• Build Understanding and Step It Out provide opportunities for students to learn and practice new mathematics, as well as “connect important processes and procedures” according to the Planning and Pacing Guide.
• Check Understanding provides a formative assessment opportunity after instruction.
• On Your Own, More Practice/Homework, Test Prep, and Spiral Review in each lesson support students in developing independent mastery of the current lessons as well as reviewing material from previous lessons.
• Lessons are in a logical order that build coherence throughout the grade level.
##### Indicator {{'3c' | indicatorName}}
There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials for HMH Into Math Grade 8 meet the expectations for having a variety in what students are asked to produce, for example:

• Performance Tasks: Unit 3, Back to the Future states, “Although time travel often occurs in movies and books, it isn’t possible in real life. But if it were possible, companies would probably exist to see trips!” This is followed by five questions which require students to use functions to represent data and find relationships.
• STEM activities: Unit 6 states, “Light travels at a speed of about 186,000 miles per second. How many miles does light travel in one hour? Write your answer in standard form and as a number multiplied by a power of ten.”
• Show written calculations and solutions
• Verbally defend or critique the work of others to show understanding
• Analyze double number lines and bar diagrams
• Build models for a problem by using diagrams and equations
• Use a diagram and a coordinate plane to represent a linear equation
• Compare multiple representations - table, graph, equation, situation - of data
• Use a digital platform to conduct and present their work
• Use manipulatives, especially in small groups, to represent mathematics
• Construct written responses to explain their thinking
##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for HMH Into Math Grade 8 present and, when appropriate, are connected to written methods.

• The series does not involve extensive use of manipulatives, however, when they are included, they are consistently aligned to the expectations and concepts in the standards.
• Most hands-on manipulatives are integrated in supplemental, small-group, differentiated instruction activities, and warm-up options.
• Examples of manipulatives include: Two-color counters, calculator, coins, number cubes, playing cards, string, square tiles, unit cubes, colored chips, algebra tiles, grid paper, index cards, anchor charts, ruler, compass, protractor, geometry software, bar diagrams, fraction strips, number lines, decimal grids, x-y tables, and pie charts.
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials for HMH Into Math Grade 8 is not distracting or chaotic and supports students in engaging thoughtfully with the subject.

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

#### Criterion 3.2: Teacher Planning

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

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for supporting teacher learning and understanding of the CCSSM. The instructional materials include: quality questions to support teachers in planning and providing effective learning experiences, a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials, a teacher edition that partially contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons, and explanations of the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials for HMH Into Math Grade 8 meet the expectations for providing quality questions to help guide students’ mathematical development.

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

Each lesson introduction poses an essential question intended to guide student learning. For example, in Lesson 13.4, the Essential Question is “How can you use volume formulas to solve problems involving cylinders, cones, and spheres?”

The Spark Your Learning planning page in the Teacher Edition includes examples of student work which show On Track, Almost There, and Common Errors. Each example has suggested questions for teachers to correct or advance student thinking. For example, in Lesson 11.1, Almost There about Pythagorean Theorem states, “What is the area of each of the smaller squares? What is the area of the larger square? Do you see a relationship among these areas?”

##### Indicator {{'3g' | indicatorName}}
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials for HMH Into Math Grade 8 meet the expectations for containing ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.

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

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

##### Indicator {{'3h' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials for HMH Into Math Grade 8 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:

• At the beginning of each module, the teacher’s edition includes Teaching for Depth that provides a brief overview of the mathematics contained in the module. For example, Module 6 states, “The set of all inputs for a function is called the domain. The set of all outputs of a function is called the range. For many functions the domain is the set of all real numbers. However, the domain is sometimes restricted to represent real-world situations. For example, in a function that describes the costs to buy tickets, the domain may be restricted to nonnegative real numbers because negative numbers of tickets do not make sense. Similarly, in an example where a function describes the time it takes to complete a task, the domain will also be restricted to nonnegative real numbers because negative quantities of time do not make sense. It is important for students to watch for reasonable domain values when working in real-world contexts. Ensure students understand that reasonable domain values must make sense.”
• In addition, Teacher to Teacher From the Classroom gives tips or anecdotes about the module content. For example, Module 3 states, “I can’t even tell you how many students I’ve seen that are well-skilled at solving linear equations. But when asked, ‘So what does that solution mean?’ they are silent. What does it mean to have a solution to an equation? What does it mean in the equation itself? What does it mean in a table of values? What does it mean in a graph? I’ve learned how critical it is for students to encounter these various representations simultaneously as they develop their understanding of mathematical ideas. Students need to connect the how of ‘isolate the variable and solve for x’ to the graph of that line and the points that are on it. This lays an important foundation for solving systems of equations down the road. My end goal is to have students who can identify solutions to linear equations as points that are on the graph of the line, coordinates in a table of values that satisfy the constraints of the equation, as well as solving for x or y algebraically.”
##### Indicator {{'3i' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The instructional materials for HMH Into Math Grade 8 meet the expectations for explaining the role of the grade-level mathematics in the context of the overall mathematics curriculum.

Each module in the Teacher Edition includes Mathematical Progressions Across the Grades which lists prior learning, current development, and future connections. Similarly, the beginning of each lesson in the Teacher Edition includes Mathematical Progressions 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 discusses the “four progressions of middle school content leading to the Algebra course: Number and Operations, Operations and Algebraic Thinking, Statistics and Probability, and Functions” and includes a table that shows how the domains in Grades 3-5, 6-7, and Grade 8/Algebra fit into these progressions.

##### Indicator {{'3j' | indicatorName}}
Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials for HMH Into Math Grade 8 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, pacing is provided in the module planning pages, and the standards contained in each lesson are identified with written descriptions as well as listed under Current Development in the Mathematical Progressions chart.

##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials for HMH Into Math Grade 8 include strategies for parents to support their students progress. The Planning and Pacing Guide describes strategies to Connect with Families and Community:

• The student materials contain Math on the Spot problems that have videos connected to them. The materials state, “Math on the Spot video tutorials provide instruction of the math concepts covered and allow for family involvement in their child’s learning.” There are generally 1-3 problems per module.
• The materials state, “Family letters inform families about the skills, strategies, and topics students are encountering at school.” Each module includes a letter, found online in four languages, providing vocabulary, a home activity, and discussion prompts.
##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The instructional materials for HMH Into Math Grade 8 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 was designed around research-based, effective teaching practices such as those described in Principles to Actions (NCTM 2014).” These include:

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

The Planning and Pacing Guide describes four design principles from the Stanford Center for Assessment, Learning, and Equity (SCALE) 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 embrace these 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.

#### Criterion 3.3: Assessment

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

The instructional materials reviewed for HMH Into Math Grade 8 partially meet expectations for offering teachers resources and tools to collect ongoing data about student progress on the CCSSM. The instructional materials provide strategies for gathering information about students’ prior knowledge, strategies for teachers to identify and address common student errors and misconceptions, and assessments that clearly denote which standards are being emphasized.

##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The instructional materials for HMH Into Math Grade 8 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, which states, “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.
##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The instructional materials for HMH Into Math Grade 8 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.
##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials for HMH Into Math Grade 6 partially meet the expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials do not provide feedback in online lessons, and in the Module Reviews.

• Each lesson ends with two or three 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.
• The online lessons are the same as in the print textbook and provide immediate notification of correct or incorrect answers, but do not provide feedback for changing incorrect answers.
• Each Module Review has a scoring guide/checklist, so students know which questions they answer correctly. The scoring guide/checklist does not provide feedback for changing incorrect answers.
• Digital assessments are auto-scored and generate recommendations that can provide feedback to teachers but not directly to students.
##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.

The instructional materials for HMH Into Math Grade 8 meet the expectations that assessments clearly denote which standards are being emphasized.

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

##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials for HMH Into Math Grade 8 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 “for immediate access to data and grouping recommendations.”
• The Planning and Pacing Guide states, “Check Understanding is a quick formative assessment in every lesson used to determine which students need additional support and which students can continue on to independent practice or challenges.” Check Understanding presents a limited number of questions, usually one to three, which includes a digital option that can be “auto-scored online for immediate access to data and recommendations for differentiation.”
• The Individual Record Forms in the Assessment Guide suggest Reteach Lessons 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 not suggest Reteach Lessons or provide other guidance that teachers can use for follow-up with students.
##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

The instructional materials for HMH Into Math Grade 8 include Scales to Track Learning Goals at the end of each lesson. The Teacher Edition introduction states, “The scales below can help you and your students understand their progress on a learning goal. Scales are also available in Module Resources.” The scale progresses from 1 to 4. For example from Grade 7, Lesson 1.1 states:

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

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

#### Criterion 3.4: Differentiation

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

The instructional materials reviewed for HMH Into Math Grade 8 meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners and strategies for meeting the needs of a range of learners. The materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations, and they provide opportunities for advanced students to investigate mathematics content at greater depth. The instructional materials also suggest support, accommodations, and modifications for English Language Learners and other special populations and provide a balanced portrayal of various demographic and personal characteristics.

##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials for HMH Into Math Grade 8 meet the expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• At the beginning of each module, Teaching for Depth provides information on strategies to use when teaching the concept, including Represent and Explain, which focuses on ways for students to describe and picture a concept, or Make Connections, which helps students understand a new idea by connecting it to previous knowledge.
• At the beginning of each module, Mathematical Progression Across the Grades makes connections to both prior and future skills and standards to scaffold instruction.
• At the beginning of each module, Diagnostic Assessment, Are You Ready? allows teachers to “diagnose prerequisite mastery, identify intervention needs, and modify or set up leveled groups.”
• Each lesson provides Warm-up Options to activate prior knowledge such as Problem of the Day, Quick Check for Homework, and Make Connections.
• Throughout the lessons, there are notes, strategies, sample guided discussion questions, and possible misconceptions that provide teachers structure in making content accessible to all learners.
• Student practice starts with up to four Check Understanding exercises to complete with guidance before moving to independent work in On My Own or More Practice/Homework.
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials for HMH Into Math Grade 8 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

• There are Reteach and Challenge activities for each lesson.
• Each module includes Plan for Differentiated Instruction that provides teachers with teacher-guided, Small-Group Options and self-directed Math Center Options based on student need: “On Track/Mixed Ability, Almost There (RtI), and Ready for More.”
• Each lesson provides Leveled Questions in the teacher’s edition identified as DOK 1, 2, and 3 with an explanation of the knowledge those questions uncover about student understanding.
• There are three “Language Routines to Develop Understanding” used throughout the materials: 1) “Three Reads: Students read a problem three times with a specific focus each time.” 2) “Stronger and Clearer Each Time: Students write their reasoning to a problem, share, explain their reasoning, listen to and respond to feedback, and then write again to refine their reasoning.” and 3) “Compare and Connect: Students listen to a partner’s solution strategy and then identify, compare, and contrast this mathematical strategy.”
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials for HMH Into Math Grade 8 meet the expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

• Each unit includes a STEM Task and a Unit Project which include multiple entry-points and a variety of solution strategies. Teachers are provided with possible answers as well as What to Watch For tips, which include “Watch for students who become discouraged by a task and quickly give up. Strategies that may help these students include: working with a supportive partner, dividing the task into smaller steps, and reminding themselves that working at a difficult task is valuable, even if the task is not completed. Taking on new challenges is how we learn” and “Watch for students who are reluctant to stretch themselves on a challenging task. Encourage these students to: identify similarities between the current task and tasks they have completed successfully in the past, identify one or more promising strategies or approaches, and try one of the strategies.”
• Each lesson begins with Spark Your Learning, which is an open-ended problem that allows students to choose their entry-point for applying mathematics and can be solved in a variety of ways. There are suggestions in the teacher’s edition to help students access the context of the problem. For example, in the side margin of the teacher’s edition, Motivate provides prompts. Grade 6, Lesson 9.1 states, “Introduce the problem. Point out that the problem does not state the amounts of money that Bella and Tia have, only that the two amounts are equal. Nevertheless, this information is enough to find a solution.” Grade 7, Lesson 1.1 states, “Introduce the problem. Ask them if they have ever used a recipe. Tell students to discuss and share with their team members in a small group.” Grade 8, Lesson 5.2 states, “Introduce the problem. Ask students if they have ever set daily goals in reading or running or some other activity. Invite students to discuss and share with their partner or team members in a small group.”
• Support for Turn and Talk in the teacher’s edition provides suggestions to help students using a variety of strategies. Teachers are often prompted to “Select students who used various strategies and have them share how they solved the problem with the class.”
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials for HMH Into Math Grade 8 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. Examples include:

• For ELLs, there is Language Development 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,” “Listen for students who do not distinguish between minus...and the negative sign,” and “Visual cues help students…”
• Language Objectives are included in every lesson.
• There are Teacher Tabletop Flipchart Activities referenced in the teacher’s edition for RtI support.
• There are Reteach, RtI Tier 2, and RtI Tier 3 worksheets that can be assigned online or printed.
• There are Turn and Talk prompts designed to support students in other special populations, such as “go back and reread the problem and break it into pieces. For example: What do you know? What do you need to find?”
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials for HMH Into Math Grade 8 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. Examples include:

• 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’s edition provides ideas for extending the task.
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials for HMH Into Math Grade 8 meet the expectations for providing a balanced portrayal of various demographic and personal characteristics. Examples include:

• Lessons contain a variety of tasks that interest students of various demographic and personal characteristics.
• Names and wording are chosen with diversity in mind. The materials include various names throughout the problems (e.g., Jayson, Suyin, Malik, Tressa, Anton, Jasmine, Yu, Felice, Sonia, Roselyn, Tracy, Tran, Arie, Miguel, Maria) that are used in ways that do not stereotype characters by gender, race, or ethnicity.
• When multiple characters are involved in a scenario, they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias, and there is no pattern in one character using more/fewer sophisticated strategies.
• When people are shown, there is a balance of demographic and personal characteristics.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for HMH Into Math Grade 8 provide opportunities for teachers to use a variety of grouping strategies. Examples include:

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

The instructional materials reviewed for HMH Into Math Grade 8 encourage teachers to draw upon home language and culture to facilitate learning. Examples include:

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

#### Criterion 3.5: Technology

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

The instructional materials reviewed for HMH Into Math Grade 8: integrate some technology in ways that engage students in the Mathematical Practices; are web-­based and compatible with multiple internet browsers; include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology; are intended to be easily customized for individual learners; and do not include technology that provides opportunities for teachers and/or students to collaborate with each other.

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

The instructional materials reviewed for HMH Into Math Grade 8 are web-based and compatible with multiple internet browsers. Examples include:

• 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. Online use was difficult on a Chromebook, there are scrolling and loading issues as well as difficulty seeing all pieces of the interactive editions.
• The materials are not compatible with an Android device (using Chrome browser). Although the website can be reached, it is not possible to zoom in or out, nor can one move the screen, so a student cannot access the entire screen.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for HMH Into Math Grade 8 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology through a website called Online ED, which parallels the print textbook. Only one module per grade is currently available in the digital format, so some of the evidence is stated in the materials but has not actually been observed. Examples include:

• Lesson problems from the Student Edition, assessments, and unit performance tasks are provided to be completed and scored using technology, providing students with feedback on whether the answers are correct or incorrect.
• Online Ed is designed to make recommendations for differentiation after auto-scoring of Check Understanding problems within each lesson.
• Growth monitoring assessments are “designed to be administered in 40 minutes, 3 times per year. The system utilizes a secure bank of assessments to adapt to each student’s ability and maps progress on the Quantile Framework.” (Pacing Guide)
• Assessments can be created using a question bank that repeats the questions presented throughout the interactive lessons. However, teachers cannot modify questions nor add new questions.
• The online system has dynamic reporting by assignment or standards. If teachers are using the online system, they can view student progress for interim growth, module readiness, and lesson practice and homework.
##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for HMH Into Math Grade 8 are intended to include opportunities for teachers to personalize learning for all students. Full functionality of online materials is not accessible at the time of this review. Examples include:

• 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 which “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 HMH Into Math Grade 8 provide minimal opportunity to be adapted for local use. Full functionality of online materials is not accessible at the time of this review. Examples include:

• 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 cannot be modified.

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

The instructional materials reviewed for HMH Into Math Grade 8 do not incorporate technology that provides opportunities for multiple students to collaborate with the teacher or one another.

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

The instructional materials reviewed for HMH Into Math Grade 8 integrate some technology, including digital lessons and virtual tools. Students can complete tasks and activities from the Student Edition through an interactive format. Examples include:

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

## Report Overview

### Summary of Alignment & Usability for HMH Into Math | Math

#### Math K-2

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

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

#### Math 3-5

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

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

#### Math 6-8

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

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

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### Overall Summary

###### Alignment
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###### Usability
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