High School - Gateway 3

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.  

The materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. The Implementation Guide includes:

• the general overview of how to use the materials,

• detailed explanation of the Math language routines,

• how to use the math prompts and the research behind them, and

• the research behind supporting diverse learners. 

The teacher materials provide the specific details of how to implement the lesson, with guidance in implementing all the components of the lesson. Examples include:

• Algebra 1, Unit 1, Lesson 1 includes warm-up 1.1 “Types of Data”, activity 1.2 “Representing Data about You and Your Classmates,” and cool down 1.3 “Categorizing Questions”. Each of these is supported by a launch, the instructional routine, and the synthesis.

• Geometry, Unit 5, Lesson 1, warm-up 1.1 ”Which One Doesn’t Belong”, activity 1.2 “Axis of Rotation”, activity 1.3 “From Three Dimensions to Two”, and cool down 1.4 “Telescope.”

• Algebra 2, Unit 1, Lesson 1 includes warm-up 1.1 “What’s Next?”, activity 1.2, “The Tower of Hanoi” activity 1.3, “Checker Jumping Puzzle,”, and cool down 1.4.

The materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes the lesson pacing, required preparation, materials, student and teacher goals, and math language routines to support the teacher. Examples include:

• Algebra 1, Lesson 1.2 states, “as students discuss their ideas, monitor for those who …”  The synthesis in the lesson provides key takeaways for the lesson as well as discussion prompts and suggested answers.

• Geometry, Unit 5, Lesson 1 Launch states, “Arrange Students in groups of 2-4. Display images for all to see.” The Activity Synthesis states, “During the discussion, ask students to explain the meaning of any terminology they use, such as round, corners, circular, or symmetric. Also press students on unsubstantiated claims.” 

• Algebra 2, Unit 1, Lesson Narrative includes both the content and practice standards as well as common misconceptions to support the standards alignment.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for  containing adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. There are supports provided for teachers to develop their own understanding of the current material being taught, and there are explanations and examples  for concepts beyond the current courses.

Lesson Narratives provide specific information about the mathematical content within the lesson and are presented in adult language. These narratives contextualize the mathematics of the lesson to build teacher understanding and give guidance on what to expect from students and important vocabulary. Examples include:

• In Algebra 1, Unit 5, Lesson 17, students investigate compounding intervals. The explanation in the 17.1 activity synthesis provides guidance to assist teachers in developing student knowledge.

• In Geometry, Unit 6, the guidance provided for teachers pertains to what students will be doing in the unit and does not extend to concepts beyond the course. “The first few lessons examine transformations in the plane. Students excounter a new coordinate transformation notation which connects transformations to functions.” The section continues in this manner relating the concepts that will be addressed throughout the unit.  

• In Algebra 2, Unit 6, Lesson 2, sample student responses and sample discussion questions and answers are provided to support teachers in teaching the material, but there are no explanations and examples beyond the current course. Questions listed for discussion include “Which side is the hypotenuse of triangle ABC and what is the length,” and “What is the sine of angle A?”

In the Implementation Guide, there are articles that contain adult-level explanations and examples of where concepts lead beyond the indicated courses. These can be used for study to renew and fortify the knowledge of secondary mathematics teachers and other educators. There are some articles that pertain to the entire series, some that pertain to courses, and some that pertain to individual units within courses. Examples include:

• In Algebra 1, Inverses is an essay from the Noyce-Dana project that pertains to the entire course. In the essay, Farrand-Shultz unifies the different contexts in which “inverses” are used in mathematics, including the multiplicative inverse of a number, the inverse of a function, and the inverse relationship between differentiation and antidifferentiation. Many of these notions can be unified by noticing the operation relative to which the inverse is defined. 

• In Geometry, there is Proof in IM's High School Geometry and Rigor in Proofs, which pertain to the entire course. In these posts, the authors, Ray-Riek and Cardone and Cardone and Rosenberg respectively situate the treatment of learning how to write proofs in IM Geometry within a structure that appears in different forms throughout a math learner’s education, including before and beyond high school mathematics.

• In Algebra 2, The Secret Life of the ax + b Group is an essay from the Noyce-Dana project that pertains to the entire course. In the essay, Howe delves into ways in which the affine group of the line (the ax + b group) surfaces in high school mathematics, and some extensions of standard topics which are suggested by this point of view.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. Correlation information is present for the mathematics standards addressed throughout the series. Each lesson has a Standards Alignment provided for the teacher. This alignment lists the standards the lesson is building on, addressing, and building towards. There is also a Standards Alignment in the Implementation Guide for all courses in the series.

Explanations of the role of the specific course-level mathematics are present in the context of the series. The lesson narrative explains the role of the course-level mathematics in most but not all lessons. Examples include:

• In Algebra 1, Unit 2, Lesson 6 students learn equivalent equations. The lesson narrative states “In middle school, students learned that two expressions are equivalent if they have the same value for all values of the variables in the expressions. They wrote equivalent expressions by applying properties of operations, combining like terms, or rewriting parts of an expression. In this lesson, students learn that equivalent …”

• In Geometry, Unit 5 has a Unit Planner that lists the alignment between the lesson and the corresponding standards. Lesson 1 lists the standards that the lesson is building on, addressing and building toward.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The materials explain the instructional approaches of the program, and reference research-based strategies for each of the instructional routines. The implementation guide includes all instructional routines at both the Unit and the Lesson levels. Appendices in the teacher editions also have references to the research. Examples include:

• Algebra 1, Unit 2, Lesson 13 utilizes the instructional routines of Math Talk, Mathematical Language Routines of Compare and Connect and Discussion Support, and Think-Pair-Share.

• Geometry, Unit 8, Lesson 1 utilizes the instructional routines of Think-Pair-Share and Which One Doesn’t Belong, as well as the Mathematical Language Routines of Stronger and Clearer, and Collect and Display.

Along with each of these routines is the research that supports the strategy. Examples include:

• The Anticipate Monitor…….Connect instructional routine states the research from “5 Practices for Orchestrating Productive Mathematical Discussions (Smith and Stein, 2011).” 

• The Language Routine Discussion Supports states the research, “To support rich discussion about mathematical ideas, representations, context, and strategies (Chapin, O'Connor, & Anderson, 2009).”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The materials include a comprehensive list of supplies needed to support the instructional activities at both the Unit and the Lesson level. The implementation guide also has a master list of all the needed materials. Examples include:

• In Algebra 1, Unit 4, the required materials are listed in the Unit Planner. The required materials are blank paper, copies of blackline master, glue or glue sticks, graphing technology, pre-printed cards cut from copies of the blackline master, pre-printed slips cut from copies of the blackline master, scientific calculators.

• In Geometry, Unit 7, the required materials are listed as colored pencils, geometry tool kits, pre-printed slips cut from copies of the blackline masters, protractors, rulers, scientific calculators, scissors, spreadsheet technology, string. 

• In Algebra 2, Unit 1, Lesson 9, the only required material is graph paper.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the content standards assessed for formal assessments, and the materials provide guidance, including the identification of specific lessons, as to how the mathematical practices can be assessed across the series.

The formal assessments include cool downs, pre diagnostics, mid unit, and end of unit for every unit in every course. There is an additional online test bank that can be searched by concept, content standard and question type. Examples include:

• The cool downs are included for every lesson in every course, and assess a standard(s) that is the focus of that particular lesson.   

• Each unit of each course has a pre diagnostic. The teacher materials list an item description for each question and when the first appearance of the skill or concept occurs in the unit, then gives advice to assist teachers when most students are struggling with the topic. The standards include below-course-level standards on which course-level standards can be built.

• Online materials list the standards in the question details for each question.

In the Implementation Guide, there is a chart for each course that identifies specific lessons within units that can be used to assess individual mathematical practices. The Implementation Guide also includes general guidance for teachers on assessing the mathematical practices, including “I can” statements for each mathematical practice that can be used to monitor students’ engagement with each practice. Examples include:

• MP1: I can show at least one attempt to investigate or solve the problem.

• MP3: I can listen to and read the work of others and offer feedback to help clarify or improve the work.

• MP4: I can refine or revise the model to more accurately describe the situation.

• MP5: I can recognize when a tool is producing an unexpected result.

• MP8: I can notice what changes and what stays the same when performing calculations, examining graphs, or interacting with geometric figures.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for including an assessment system that provides multiple opportunities throughout the courses and series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. All assessments provide a possible rationale that also includes misconceptions to support students who choose those answers. Examples include:

• Rubrics are provided for short answer, constructed response, and extended response prompts on the mid-unit assessments and end-of-unit assessments. The implementation guide explains a three tier classification for constructed response. Extended response items have a 4 tier classification. Answers are included showing how students could get a correct answer and/or what the student could have done wrong. Sample errors and acceptable errors are provided to specify exclusions.

  • For example, item 7 on the Algebra 2 Unit 4 end-of-unit assessment provides a sample student response and rubric describing the 4 tier classification. To earn a Tier 1 classification “work is complete and correct, with complete explanation or justification” and gives a sample “1. (4.5, 310)(or comparable values); the two molds cover the same area at this time. 2. Use the equation $$100 \cdot e^{0.05d} = 100$$ and solves correctly with the natural logarithm.”

  • The three tier classification for constructed response has a general rubric stating, “Tier 1: work is complete and correct, Tier 2: Work shows general conceptual understanding and mastery with some errors, Tier 3: Significant errors in work demonstrate lack of conception understanding and mastery.”

• Rubrics are provided for the modeling prompts and include sample student responses.  All modeling tasks are assessed on two skills: “use your model to reach a conclusion” and “refine and share your model”. Suggestions are provided for how students can improve their score for each skill. 

• All Multiple Choice/Multiple Response options include an item analysis noting any misconceptions and why students may be selecting the wrong answers. 

• The pre-unit diagnostic for Algebra 1, Unit 4, Question 1 states, “If most students struggle with this item: Plan to spend additional time in Lesson 1, Activity 2 calculating how to scale the passport photo by different percentages and if additional practice is needed after Lesson 1 to connect growth factor and percent change, revisit Unit 1, Lesson 19, Activity 1, or use the optional Lesson 2 for more support.”

• The pre-unit diagnostic for Geometry, Unit 4, Question 2 states, “First Appearance of the skill: Lesson 1 Item Description: In a previous unit, students proved the Pythagorean Theorem and used it to find lengths of sides of right triangles. This item will assess whether they may compute an irrational length that needs to be rounded. If most students struggle with this item plan to invite students to find the length of the ramp in Activity 3 in addition to stating the dimensions of the legs.”

• An online customizable question bank allows the teacher to target specific objectives and standards.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. Examples include:

• In Algebra 1, Unit 7, End-of-Unit Assessment, Item 2, students engage in MP7 when they identify all expressions equivalent to $$x^2 + 6x = 16$$. 

• In Geometry, Unit 2, End-of-Unit Assessment, students have a Multiple Response item to select all statements that are true about a parallelogram. There are 2 Multiple Choice items about similar triangles, and 4 items that are short answer or extended response items about proofs of congruent segments in a parallelogram, measurements of angles of a parallelogram, measurements of an isosceles triangle and rigid motions in proving figures are congruent.

• In Geometry, Unit 3, End-of-Unit Assessment, Item 6, students engage in MP’s 5 and 6 when they reason through “Tyler’s” proof and find his mistakes. 

• In Algebra 2, Unit 2, Mid-Unit Assessment, Item 5, students engage in MP2 when they find a coefficient of a polynomial, a factor of the polynomial, and explain how they know this is the correct coefficient.

• In Algebra 2, Unit 6, End-of-Unit Assessment, students write an equation and sketch a graph for another item on the assessment, and students explain what the different parameters of the equation mean in context for another assessment item.

• All lessons across the series include cool downs at the end of the lesson, to allow students to show they understood the work of that day’s lesson.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics. Across the series, every lesson has supports and resources provided to teachers to help engage and support students with disabilities in course-level mathematics, as well as students in special populations. Examples include:

• In Algebra 1, Unit 4, Lesson 1, "Three Reads" is provided to support students reading comprehension. The teachers notes state to, “use the first read to orient students to the situation. After a shared reading, ask students ‘what is this situation about?’” and goes on to provide support for the teacher in two additional reads with targeted student outcomes.

• In Geometry, Unit 1, Lesson 6, Math Talk, there are discussion supports that provide sentence frames to support students in explaining their strategies. Teachers are encouraged to have students share their answers with a partner, rehearsing what they will say in large group discussions. The teacher notes clarify that rehearsing provides “opportunities to clarify their thinking.”

• In Geometry, Unit 1, Lesson 8, “Digital Compass and Straightedge Constructions,” encourages teachers to keep the construction on display for students to reference as they work to support an accessibility for “Memory: Conceptual Processing.”

• In Algebra 2, Unit 5, Lesson 8, teachers are encouraged to “represent the same information through different modalities by drawing a diagram. Encourage students who are unsure where to begin to sketch a diagram of a slice of break on graph paper and to share the area that is covered in mold after 1 day, after 2 days, …”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing extensions and opportunities for advanced students to engage with course-level mathematics at higher levels of complexity. The materials provide multiple opportunities for all students to investigate course-level content at higher levels of complexity. Most lessons throughout the series have Are You Ready For More?. The Implementation Guide indicates that Are You Ready For More? problems are differentiated for students who are ready for more of a challenge. There are also optional lessons and optional activities in some lessons that may support learning at a higher level of complexity. If individual students would complete these optional activities, then they might be doing more assignments than their classmates. Examples of these activities include:

• In Are You Ready For More?, Algebra 1, Unit 4,  Lesson 11, students describe and graph unique functions given specific limitations on the domain and/or range.

• In Are You Ready For More?, Geometry, Unit 7, Lesson 9, students analyze the relationships between a sector and the full circle. 

• In the Implementation Guide, “The reason an activity is optional is that it addresses a skill or concept below grade level or addresses a concept or skill that goes beyond the requirements of the standard, or provides an opportunity for additional practice on a concept or skill.”  For example, in Geometry, Unit 7, Lesson 14, there are optional activities 14.3, “A Fair Split”, and 14.4, “Let Your Light Shine.” Students use what they know about trigonometry and circles to decide how to divide a pizza slice between two people equally so that one of them doesn’t have to eat the crust.

There is no clear guidance for the teacher on ways to specifically engage advanced students in investigating the mathematics content at higher levels of complexity.

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning course-level mathematics. The implementation guide states, “The framework for supporting English Language Learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” Examples include:

• In Algebra 1, Unit 6, Lesson 5, teacher guidance states “Speaking, Reading: MLR5 Co-Craft Questions. Begin the launch by displaying only the context and the diagram of the building. Give students 1–2 minutes to write their own mathematical questions about the situation before inviting 3–4 students to share their questions with the whole-class. Listen for and amplify any questions involving the relationship between elapsed time and the distance that a falling object travels. This routine meets the Design Principle(s): maximize meta-awareness and cultivate conversation.”

• In Geometry, Unit 6, Lesson 14, there is a Collect and Display language routine that directs teachers to note, “As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments BD and CD is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, ‘The lines make 90 degrees because they have opposite slopes’ can be improved by restating it as ‘Segments BD and CD are perpendicular because their slopes are opposite reciprocals.’ This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments BD and CD is 90 degrees.”

• In Algebra 2, Unit 4, Lesson 14, there is a Compare and Connect language routine that states, “Use this routine to prepare students for the whole-class discussion about the strategies for finding the year when the population of Country C reached 30 million. After students find the year when the population reached 30 million, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the strategies for finding the value of t so that f(t) = 30. This will help students understand and find connections between multiple approaches to solving this problem. Design Principle(s): Cultivate conversation.”

The materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Guidance is provided for both online and physical manipulatives. The online materials have access to any digital manipulatives needed, without using any physical manipulatives.

When students use the interactive student edition, they are provided with various digital tools including a drawing tool, a compass and protractor, a ruler, a statistics tool, and a generic graphing calculator. Examples include:

• In Algebra 1, Unit 1, Lesson 9, interactive student edition, students create a digital dot plot, a box plot, and a histogram via an embedded statistics tool to create models and graphical representations.

• In Geometry, Unit 3, Lesson 1, students use the drawing tool to dilate lines. 

• In Geometry Unit 4, Lesson 4, students use a web applet and a slider to change a right triangle and observe how the side lengths and angle measures change, then compare the ratio of side lengths of different triangles.

Using the physical materials, students use graph paper, graphing/scientific/four-function calculators, bouncing balls, rulers, cubes, colored pencils, blackline masters, measuring tapes, markers, clay, dental floss, pasta, isometric dot paper, mirrors, etc. Examples include:

• In Algebra 1, Unit 2, Lesson 12, students use graphing technology to model various situations in developing the meaning of the solution to a system of equations.

• In Geometry, Unit 2, Lesson 4, students use linguini pasta to support proving that two triangles are congruent. 

• In Geometry, Unit 7, Lesson 2, students use manipulatives both online and physically to measure angles and describe the relationship between an inscribed angle and the central angle that define the same arc.