High School - Gateway 3

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson. It is clear when students solve problems to learn and when they apply skills.

Lessons include a Warm-Up, Activities, Synthesis, and a Cool-Down. Practice Problems are in a separate section of the instructional materials, distinguishing between problems students complete and exercises in the lessons. Warm-Ups connect prior learning or engage students for learning new material in the lesson. Students learn and practice new mathematics in lesson Activities. In the Synthesis activity, students build on their understanding of the new concept. Each activity lesson ends with a Cool-Down in which students apply what they have learned from the activities, complete preliminary practice, or complete an introduction to skills they may need in the next lesson.

Practice problems are consistently found in the Practice sets that accompany each lesson. These sets of problems include problems that support students in developing mastery of the current lesson and unit concepts and review of material from previous units. When practice problems contain content from previous lessons, students apply their skills and understandings in different ways that enhance understanding or application (e.g., increased expectations for fluency, more abstract application, or a non-routine problem).

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for not being haphazard; exercises are given in intentional sequences.

Overall, clusters of lessons within units and activities within lessons are intentionally sequenced so students develop understanding. The structure of a lesson provides students with the opportunity to activate prior learning, build procedural skill and fluency, and engage with multiple activities that are sequenced from concrete to abstract or increase in complexity. Lessons end with a Cool-Down which is aligned to the daily lesson objective. Unit sequences consistently follow progressions to support students' development of conceptual understanding and procedural skills.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for having variety in what students are asked to produce.

The instructional materials prompt students to produce products in a variety of ways. Students produce solutions within Activities and Practice, as well as participating in class, groups, and partner discussions. Materials provide opportunities for students to construct viable arguments and critique the reasoning of their peers. Students use a digital platform and paper-pencil to conduct and present their work. The materials consistently prompt students for solutions that represent the language and intent of the standards. Students use representations such as tables, number lines, area diagrams, dot plots, geometric constructions, and graphs, as well as strategically choose tools to complete their work (MP5). Lesson activities and tasks are varied within and across lessons.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The series includes a variety of virtual manipulatives and integrates hands-on activities that allow the use of physical manipulatives, for example:

• Manipulatives and other mathematical representations are consistently aligned to the expectations and concepts in the standards. The majority of manipulatives used are commonly accessible measurement and geometry tools.
• The materials provide digital applets for manipulating geometric shapes, such as GeoGebra applets, tailored to the lesson content and tasks. When physical, pictorial, or virtual manipulatives are used, they are aligned to the mathematical concepts they represent. For example, in Geometry, Lesson 5, Activity 10.2, two rectangular prisms with the same base area and the same height are used within an applet to develop Cavilieri’s Principle.
• Examples of manipulatives for Geometry include: an index card to use as a straightedge, compasses, tracing paper, blank paper, colored pencils, and scissors. Additionally, GeoGebra applets are used for constructions, to perform transformations, to explore congruence and similarity, and to visualize cross sections.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet the expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson consists of a detailed lesson plan accompanied by teaching notes. Included in these teaching notes are the objectives of the lesson, suggested questions for discussion, and guiding questions designed to increase classroom discourse and foster understanding of the concepts. For example, in Geometry, Unit 4, Lesson 6.2 suggests the teacher ask, “The right triangle table is useful, but what if the angle is not a multiple of 10 degrees?” In Algebra 1, Unit 3, Lesson 7, Lesson Synthesis, the curriculum suggests the following question for discussion, “What does a scatter plot look like when its line of best fit has a correlation coefficient of -0.5? Sketch it.” The teaching notes and questions for discussion support the teachers in planning and implementing lessons effectively.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet the expectations for providing teacher supports with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

• Each lesson includes the Learning Goals written for teachers and students, learning targets written for students, a list of Word/PDF documents that can be downloaded, CCSSM Standards that are “built upon” or “addressed” for the lesson, and any instructional routines to be implemented. Within the technology, there are expandable links to standards and instructional routines.
• Lessons include detailed guidance for teachers for the Warm-Up, Activities, and the Lesson Synthesis.
• Each lesson activity contains an Overview and Launch Narrative, Guidance for Teachers and Student-facing materials, Anticipated Misconceptions, “Are you ready for more?”, and an Activity Synthesis. Included within these narratives are guiding questions and additional support for students.
• The teacher materials that correspond to the student lessons provide annotations and suggestions on how to present the content. “Launch” explains how to set up the activity and what to tell students. After the activity is complete, there are often Anticipated Misconceptions in the teaching notes, which describes how students may incorrectly interpret or misunderstand concepts and includes suggestions for addressing those misconceptions.
• The materials are available in both print and digital forms. The digital format has embedded GeoGebra applets. Guidance is provided to both the teacher and the student on how to use the Geometry Toolkit and applet. For example, in Geometry, Unit 1, Lesson 1, teachers and students are provided access and time to play with the applet tools. During the Launch, teachers are encouraged to model the different tools, practice with the students, and answer questions.

The instructional materials reviewed for the Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for the teacher’s edition containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.

The narratives provided for each unit include information about the mathematical connections of concepts being taught. Previous and future grade levels are also referenced to show the progression of mathematics over time. Important vocabulary is included when it relates to the “big picture” of the unit.

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.

The Lesson Narrative for Algebra 2, Unit 3 states, “In the next set of lessons, students connect the $$\sqrt{ }$$ and $$\sqrt[3]{ }$$ symbols with solutions to quadratic and cubic equations. Students learn that a number is a square root of c if it squares to make c. In other words, square roots of c are solutions to the equation $$x^2=c$$. Students use the graph of $$y=x^2$$ to see that all positive numbers have two square roots, one positive and one negative. They learn the convention that the positive square root is given the symbol $$\sqrt{ }$$, so the positive square root of c is written $$\sqrt{c}$$ and the negative square root is written $$-\sqrt{c}$$.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Course Guide and Narratives describe how mathematical concepts are built from previous grade-level/course and lesson material. For example, in Algebra 1, Unit 4, the Lesson Narrative states, “In grade 8, students learned that a function is a rule that assigns exactly one output to each input. They represented functions in different ways—with verbal descriptions, algebraic expressions, graphs, and tables—and used functions to model relationships between quantities, linear relationships in particular.” In addition, in Algebra 1, Unit 5, the Lesson Narrative states, “Before starting this unit, students are familiar with linear functions from previous units in this course and from work in grade 8. They have been formally introduced to functions and function notation and have explored the behaviors and traits of both linear and non-linear functions. Additionally, students have spent significant time graphing, interpreting graphs, and exploring how to compare the graphs of two linear functions to each other. In this unit, students frequently use the properties of exponents, a topic developed in grade 8. They also apply their understanding of percent change from grade 7 and use an exponent to express repeated increase or decrease by the same percentage.”

For some units, there are explanations given for how the grade-level concepts fit into future high school work. For example, in Geometry, Unit 7, the Lesson Narrative states, “Students develop fluency with radian measures by shading portions of circles and working with a double number line. This is important for the transition towards Algebra 2. In that course, students will explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels.

• Prior grade-level or course standards are indicated in the instructional materials. The lesson Warm-Up is designed to engage students' thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills.
• Prior knowledge is gathered about students through the pre-unit Check Your Readiness assessments. In these assessments prerequisite skills necessary for understanding the topics in the unit are assessed. Commentary for each question provides the relevance of the questions to the topic and a list of standards assessed is provided for the teacher. For example, in Algebra 2, Unit 5, Check Your Readiness assessment problem 2 shows 7.G.1 as an aligned standard. The teacher note states, “Geometry Unit 3 Lesson 1 Activity 4, Match the Scale Factors, is an example of a brief activity that could be added to review scale factors.” The notes also state, “Vertical and horizontal stretches are not dilations because the stretch is only applied in a single direction while a dilation applies the same scale factor in all directions. Nonetheless, the idea of scaling is common to both situations.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

 Lesson Activities include teaching notes that identify where students may make a mistake or struggle. There is a rationale that explains why the mistake could have been made, suggestions for teachers to make instructional adjustments for students, and steps teachers can take to help clear up the misconceptions. For example, in Geometry, Unit 2, Lesson 6.2, the teacher notes state, “Anticipated Misconceptions: If students are searching too far back, point students toward the proof in the warm-up activity, Information Overload. The goal is for students to understand and adapt that proof to this situation, so help students find the proof relatively quickly so they can have time to engage in productive struggle as they try to understand and adapt it.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

 The lesson structure consisting of a Warm-up, Activities, Lesson Synthesis, and Cool-down provide students with opportunities to connect prior knowledge to new learning, engage with content, and synthesize their learning. Throughout the lesson, students have opportunities to work independently, with partners, and in groups where review, practice, and feedback are embedded into the instructional routine. Practice Problems for each lesson activity reinforce learning concepts and skills and enable students to engage with the content and receive timely feedback. In addition, discussion prompts provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for assessments clearly denoting which standards are being emphasized.

 Assessments are accessed through the Assessment tab for each unit and are available in two print options. For each unit, there is a Check Your Readiness and an End-Unit Assessment. Longer units also include a Mid-Unit Assessment. Assessments begin with guidance for teachers on each problem, followed by the student-facing problem, solution(s), and the standard targeted.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Assessments include an answer key, and when applicable, a rubric consisting of three to four tiers, ranging from Tier 1 (work is complete, acceptable errors) to Tiers 3 and 4 (significant errors, conceptual mistakes).

Assessments include multiple choice, multiple response, short answer, restricted constructed response, and extended response. Restricted constructed response and extended response items have rubrics that are provided to evaluate the level of student responses. The restricted constructed response includes a 3-tier rubric, and the extended constructed response includes a 4-tier rubric. For these types of questions, the teacher materials provide guidance as to what is needed for each tier as well as some sample responses.

In the Assessment Teacher Guide for each End of Unit Assessment, there are narratives about what may have caused students to choose an incorrect response before the problems are shown along with the correct responses and aligned standards. For example, in Algebra 2, Unit 5, End of Unit Assessment, Problem 2, the Assessment Teacher Guide states, “Students who select A may be confusing how to represent horizontal translations using function notation. Students who select C need more work with representing shifts, reflections, and stretches using function notation. Students who select D may be confusing how to represent horizontal and vertical reflections using function notation.”

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson is designed with a Warm-Up that reviews prior knowledge and/or prepares all students for the activities that follow, and the Cool-Down reviews the concepts of the lesson.
• Within a lesson, narratives provide explicit instructional support for the teacher, including the Activity Launch, Anticipated Misconceptions, and Lesson Synthesis. This information assists teachers in making the content accessible to all learners.
• Lesson Narratives often include guidance on where to focus questions in Activities or the Lesson Synthesis.
• Optional activities are often included that can be used for additional practice or support before moving to the next activity or lesson.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

The lesson structure—Warm-Up, Activities, Lesson Synthesis, and Cool-Down—includes guidance for the teacher on the mathematics of the lesson, possible misconceptions, and specific strategies to address the needs of a range of learners. Embedded supports include:

• Mathematical Language Routines to support a range of learners to be successful are provided for the teacher throughout lessons to maximize output and cultivate conversation. For example:
• MLR1: Stronger and Clearer Each Time, in which “students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response.”
• MLR4: Information Gap, which “allows teachers to facilitate meaningful interactions by giving partners or team members different pieces of necessary information that must be used together to solve a problem or play a game...[S]tudents need to orally (and/or visually) share their ideas and information in order to bridge the gap.”
• MLR6: Three Reads, in order to "ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented. This routine supports reading comprehension of problems and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner in mathematical conversation.”
• Teaching notes appear frequently in lessons to provide additional guidance for teachers on how to adapt lessons for all learners. These teaching notes state specific needs addressed in a recommended strategy that is relevant to the given task and includes supports for Conceptual Processing, Expressive & Receptive Language, Visual-Spatial Processing, Executive Functioning, Memory, Social-Emotional Functioning, and Fine-motor Skills. For each support, there are multiple strategies teachers can employ, for example: Conceptual Processing includes strategies to Eliminate Barriers, Processing Time, Peer Tutors, Assistive Technology, Visual Aids, Graphic Organizers, and Brain Breaks.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for embedding tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

The problem-based design engages students with complex tasks multiple times each lesson. The Warm-Up, Activities, Lesson Synthesis, and Cool-Down provide opportunities for students to apply mathematics from multiple entry points.

Specific examples of strategies found in the materials include “Notice and Wonder” and “Which One Doesn’t Belong?” The lesson and task narratives provided for teachers offer possible solution paths and presentation strategies from various levels, for example:

• In Geometry, Unit 6, Lesson 7, students consider a set of points that are equidistant from a line and a given point. Students share what they notice and what they wonder. This takes place in the Warm-Up of this lesson, and offers an opportunity for all students to access the content.
• In Algebra 1, Unit 4, Lesson 8, students compare four graphs of temperature over time. Students may realize that each of the four representations might not belong based on different criteria. This instructional routine allows students to be precise in their language and to define the parameters necessary when solving contextual problems.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for including support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The ELL Design is highlighted in the Teacher Guide and embodies the Understanding Language/SCALE Framework from the Stanford University’s Graduate School of Education, which consists of four principles: Support Sense-Making, Optimize Outputs, Cultivate Conversation, and Maximize Meta-Awareness. In addition, there are eight Mathematical Language Routines (MLR) that were included “because they are the most effective and practical for simultaneously learning mathematical practices, content, and language.” "A Mathematical Language Routine refers to a structured but adaptable format for amplifying, accessing, and developing students’ language."

ELL Enhanced Lessons are identified in the Unit Overview. These lessons highlight specific strategies for students who have a language barrier which affects their ability to participate in a given task. Throughout lessons, a variety of instructional routines are designed to assist students in developing full understanding of math concepts and terminology. These Mathematical Language Routines include:

• MLR2, Collect and Display, in which “The teacher listens for, and scribes, the student output using written words, diagrams and pictures; this collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use.”
• MLR5, Co-Craft Questions and Problems, which “[allows] students to get inside of a context before feeling pressure to produce answers, and to create space for students to produce the language of mathematical questions themselves.”
• MLR7, Compare and Connect, which “[fosters] students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language.”

Lesson Narratives include strategies designed to assist other special populations of students in completing specific tasks. Examples of these supports for students with disabilities include:

• Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.
• Conceptual Processing: Eliminate Barriers. Assist students in seeing the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.
• Conceptual Processing: Processing Time. Check in with individual students as needed to assess for comprehension during each step of the activity.
• Executive Functioning: Graphic Organizers. Provide a t-chart for students to record what they notice and wonder prior to being expected to share these ideas with others.
• Memory: Processing Time. Provide students with a number line that includes rational numbers.
• Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics Traditional series meet expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

All students complete the same lessons and activities; however, there are some optional lessons and activities that a teacher may choose to implement with students. For example, in Algebra 1, Unit 3, Lesson 10 is an optional lesson intended to provide an opportunity for students to use skills and knowledge gained from other lessons in this unit.

“Are you ready for more?” is included in some lessons to provide students additional interactions with the key concepts of the lesson. Some of these tasks would be considered investigations at greater depth, while others are additional practice.

There is no clear guidance for the teacher on ways to specifically engage advanced students in investigating the mathematics content at greater depth.