High School - Gateway 3

The instructional materials reviewed for the Discovering series meet expectations that the underlying design of the materials distinguishes between problems and exercises. Each lesson presents problems designed to introduce new content along with investigations and explorations. After new content is introduced, students complete Exercises designed to engage them in content of the lesson. Many lessons include a performance task in the Exercises. The lesson resources also provide Launch problems, One-Step problems, Extra Examples, Closing Questions, and Developing Proof.  

In Discovering Algebra, there are Dynamic Explorations which are designed to help students develop knowledge of the content in the chapter, and these explorations are aligned to specific lessons. In Discovering Geometry, there are Dynamic Math Investigations, where students use the sketch program to investigate concepts in each chapter. These dynamic explorations/investigations enable students to develop knowledge from problems as opposed to practicing and applying knowledge in exercises. In Discovering Advanced Algebra, there are Explorations which are similar to the Investigations found in the Student and Teacher Editions. These Explorations allow students to further develop knowledge of concepts from problems rather than exercises. The Explorations found in Discovering Advanced Algebra are not linked to any particular chapter or lesson.

In all three products, there are More Practice Your Skills (in Discovering Geometry, these are titled Practice Your Skills), which are more exercises for students to practice the content they have learned.

The instructional materials reviewed for the Discovering series meet expectations that the design of the assignments is not haphazard and given in intentional sequences. The materials present problems for students to explore content initially and then exercises for students to practice the content. Students complete a launch problem to have productive struggle with the content and determine their prior knowledge. In One-Step problems, students further grapple with the content. Students develop understanding of the content through problems, investigations, and explorations within the lesson. Students then complete exercises to practice the new content. The teacher materials provide ample additional resources that include content-rich tasks that could be provided to students.

Concepts build upon each other within the courses. For example, in Discovering Algebra, Lesson 5.3 includes linear equations written in different forms, and it builds upon work in Lesson 5.1, where linear equation are expressed as y in terms of x. In Discovering Geometry, students encounter a variety of Dynamic Explorations placed within lesson presentations and exercises to allow students to explore concepts and develop understanding. For example, in Lesson 4.3, students use the Dynamic Exploration as a visual representation of triangle inequalities and the exterior angle theorem.

The One-Step problems offer ways to modify the investigations (present problems from the lesson in different ways), so there is not as much scaffolding. For example, in Discovering Advanced Algebra, Lesson 4.3, the One-Step problem presents Example B in only one question where Example B in the lesson uses three parts to scaffold the students’ answer to the ultimate question “How loud was the bell when it was struck?”

The instructional materials reviewed for the Discovering series meet expectations that there is variety in how students are asked to present the mathematics. In Discovering Geometry, students solve problems by creating sketches, using dynamic geometric software, completing flowmaps, constructing proofs, and writing descriptions and explanations of their work. In Discovering Algebra and Discovering Advanced Algebra, methods for presenting mathematics include creating graphs, tables, equations, descriptions, and reports.

The variety of solutions required for the problems extends to the Assessment Resources provided with each course. The Modeling Tasks also add some variety to the types of responses required throughout the series. In the Cohesive Assessment System, teachers may select exercises that require constructed responses (called Essay questions in the system), Subjective Short Answer questions requiring short responses, and multiple-choice questions with one correct answer.

The instructional materials reviewed for the Discovering series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. Students work with virtual manipulatives and through explorations using physical objects as directed in the lessons. In Discovering Algebra, students roll a pencil down a book at a given angle and collect data about the the distance the pencil travels. These manipulatives assist students in understanding the concepts as students create a model from the exploration. Students use coordinate grids and graph paper to construct polygons in Discovering Geometry, as well as interactive graphing software for explorations.

In Discovering Advanced Algebra, students make algebraic connections primarily using equations, tables, and graphs. Also in Discovering Advanced Algebra, an Exploration on Geometric Probability is provided in the digital product resources. In this exploration, students use a ruler, penny, nickel, dime, quarter, and grid paper to investigate a problem. The coins are used to represent the different sizes of coins. The grid paper is used as a representation of congruent tiles mentioned in the problem.

The instructional materials reviewed for the Discovering series meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. In the Teacher Edition under Teaching the Lesson, teachers are provided questions to guide students’ mathematical understanding and detailed support for how to teach each lesson. Teachers are also provided Critical Questions to ask students about the Big Idea of the activity. For example, in Discovering Geometry, Lesson 6.1, the Critical Question is “Is a rotation through 0° a rotation?”

The Critical Questions and Big Ideas are placed throughout the lesson in the teacher’s notes to help the teacher check for student understanding at different points in the lesson. In addition to the Critical Questions and Big Ideas, the materials provide questions labeled Ask in the teacher notes. For example, in Discovering Advanced Algebra, Lesson 4.1, the teacher notes for Example B include “Why are these functions equivalent?” along with an explanation on how to answer the question. In Summarize in the same lesson, teachers are presented with three additional questions for students to consider on increasing exponential functions and how they relate to the work with half-life completed in the lesson.

The instructional materials reviewed for the Discovering series meet expectations for containing a Teacher Edition with ample and useful annotations and suggestions on how to present the content in the student edition and ancillary materials. Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. The materials include useful annotations and suggestions on how to teach the content in Teaching the Lesson in the Teacher Edition for each lesson. The materials include step-by-step instructions about how to Launch the lesson and conduct the investigations and explorations. Teachers are also provided formative assessment instructions as well as ways to summarize each lesson. The materials provide teacher guides for lessons in the ancillary materials. Teachers also have detailed notes on how to use the TI-84 and Ti-Nspire during lessons, and support is provided around the interactive investigations in the Discovering Geometry ancillary materials.

In the Teachers Notes for each section, there are Objectives, Vocabulary, Materials needed for the lesson, and connections to math practices. There is also a brief explanation that gives an overview of the content to be addressed in the lesson, and in some cases, there are connections to previous lessons. Additionally, teachers are provided extra examples in Apply of the Teacher Notes along with Closing Questions and suggested exercises for students in Assigning Exercises. In the Teacher Notes, teachers are given explanations for what to watch for as students complete the problems. This may include questions to ask for various exercises that can help to deepen the understanding of the exercise, information about what is included in the exercise (in Discovering Advanced Algebra, Lesson 4.1 Exercise 8, “...the data have decimal values in the exponent.”), and alerts for possible misconceptions or errors.

The instructional materials reviewed for the Discovering series meet expectations for containing a Teacher Edition that contains full, adult-level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary. The teacher ebook for each course contains guidance within Teaching the Lesson and Answer for each lesson. One tab of Teaching the Lesson and Answer provides The Mathematics at the beginning of the unit, and The Mathematics provides an overview of the terms within the unit and their connection to the content within the course and beyond.

Discovering Algebra and Discovering Advanced Algebra supplemental materials also include Problem Strings. Problem Strings provides guidance for both the student and teachers related to concepts for each unit. For example, the problem string for Discovering Advanced Algebra, Lesson 6.1, provides information that “the strategy we are calling ‘adding ordinates’ comes from the (abscissa, ordinate) language of ordered pairs. It simply means adding the y-values from the parts of a combination function.”

The instructional materials reviewed for the Discovering series meet expectations for gathering information about students' prior knowledge within and across grade levels/courses. The materials provide opportunities for teachers to access students' prior knowledge in the Launch at the beginning of each lesson. For example, in Discovering Geometry, Lesson 8.1, “A rectangle has dimensions 6 ft by 8 ft. What is the area and perimeter of the rectangle? Describe what the perimeter and area each represent.” In Discovering Algebra, each chapter includes Refreshing Your Skills, which includes exercises related to topics from previous courses that will be used in the chapter.

The instructional materials reviewed for the Discovering series meet expectations for providing support for teachers to identify and address common student errors and misconceptions. In Teaching the Lesson in the teacher ebook, the materials provide common misconceptions to problems in the exercises and advice on how to further student understanding if these misconceptions occur. For example, in Discovering Geometry, Lesson 4.7, “Students may claim that they don’t have enough information to prove the given triangles congruent. Ask “Can you make any triangles that you could prove congruent from the given information?” Teachers also look for misconceptions during group activities and are provided support on how to address these misconceptions. For example, in Discovering Algebra, Lesson 8.2, “Note any groups that plot (width, length) instead of (width, area). As students describe their graphs and whether the points should be connected, ask whether others in the class agree or disagree and why.” In Discovering Advanced Algebra, Lesson 6.2, students investigate maximizing volume in boxes, and in Step 7, teachers are given information about how to address students’ understanding of maximum based on the real-world domain for the function they are using. This clarifies misunderstandings regarding maximum and minimum values for polynomial functions.

The instructional materials reviewed for the Discovering series meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. In Teaching the Lesson provided in the teacher ebook, the materials provide feedback to give to students throughout the lesson and during the exercises on both concepts and skills. For example, in Discovering Advanced Algebra, Lesson 6.3, the materials give the following guidance, “To expand these expressions, students can either draw rectangle diagrams or distribute systematically. Support: To multiply three or more factors, suggest that students use the problem-solving technique of breaking down the problem into smaller problems.” The materials also provide suggested student feedback in the supplementary materials. The Discovering series provides ongoing review and practice with a review at the end of each chapter that includes additional exercises and has the same type of feedback provided for teachers that was provided throughout the lessons. In the ancillary materials, there are More Practice Your Skills that offer additional practice exercises for each lesson.

The instructional materials reviewed for the Discovering series meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide scaffolding for the majority of problems, and the materials are differentiated for groups of students. In the Teacher Edition, there is a tab that includes Modifying Investigation, Conceptual Procedural, Differentiated Instruction, and CCSS. Modifying Investigation includes ways the teacher could modify the investigation for time or levels of support. Conceptual Procedural describes how the lesson addresses conceptual and/or procedural teaching/learning. Differentiated Instruction includes strategies/statements for addressing content for ELL students, how to offer additional support for the students who need it, and how to modify the assignment/lesson for advanced learners. The suggested adjustments to instruction are not described in depth, and the materials do not provide justifications for the recommended adjustments to instruction.

The instructional materials reviewed for the Discovering series meet expectations for providing teachers with strategies for meeting the needs of a range of learners. General strategies to meet the needs of all learners are included in Teaching the Lesson and in the Guide for Teachers in the supplemental materials. The general strategies suggested by the materials include ways to incorporate student grouping and exploration activities. In Teaching the Lesson, there are strategies labeled as Support, ELL, Language, and Alert that help the teacher make the materials accessible to all students. For example, in Discovering Advanced Algebra, Lesson 3.5, Teaching the Lesson gives Support, “If students are having difficulty reflecting the entire graph at once, encourage them to reflect each of the four marked points separately before reconnecting the segments. Students could use tracing paper to trace the function y = f(x) and perform the reflection in one step.”

The instructional materials reviewed for the Discovering series meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. There are several problems with multiple entry-points or problems with multiple solutions or representations. These problems are found within each lesson during explorations and within the modeling tasks included in the supplementary materials.

In the Guide for Teachers, the materials state that there is flexibility by allowing teachers to choose which investigations to use and which investigations they might omit. Teachers also have the flexibility to use the One-Step problem in lieu of the full scaffolded investigation for most lessons, which can offer more opportunities for multiple entry-points and/or correct responses. There are also Projects included for each course. For Discovering Algebra, these Projects are embedded in the lessons in the student and teacher ebooks. For Discovering Geometry and Discovering Advanced Algebra, these Projects are included in the ancillary, digital materials and referenced in Teaching the Lesson for each chapter. These Projects offer additional opportunities for problems with multiple entry-points and multiple correct responses.