8th Grade - Gateway 2

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)

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)

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)

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.

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.”

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.”

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.”

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.”