High School - Gateway 3

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having an underlying design that distinguishes between problems and exercises.

• Each Topic includes three sections: Overview, Exploring, and Summary. The Overview section introduces the mathematical concepts that will be addressed in the Topic. The Exploring section includes two to four explorations. In these explorations, students learn the mathematical concepts of the Topic through problems that include technology-enhanced animations and full-class activities. The Summary section highlights the most important concepts from the Topic and gives students another opportunity to connect these concepts with each other.
• Each Topic also includes three additional sections: Practice, Assessment, and Activity Sheets. The Practice section includes Guided Practice and More Practice. Guided Practice consists of exercises that students complete during class periods, providing opportunities for students to apply the concepts learned during the explorations. More Practice contains exercises that are completed as homework assignments. The Assessment section includes Automatically Scored and Constructed Response. These items are exercises to be completed during class periods or as part of homework assignments. They provide more opportunities for students to apply the concepts learned during the explorations. The Activity Sheets also contain exercises, which can be completed during class periods or as part of homework assignments, that are opportunities for students to apply the concepts learned during the explorations.
• Some Topics also include MARS Tasks, which are exercises that present students with opportunities to apply concepts they have learned from the Topic in which the MARS Task resides or to apply and connect concepts from multiple Topics.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences.

The sequencing of Topics, and explorations within the Exploring section for each Topic, develops in a way that helps to build students’ mathematical foundations.

• The Topics are comprised of similar content. For example, in Algebra I, Topic 3 Functions, the Exploring section consists of: Function Notation, Modeling with Functions, and Graphs.
• Within the explorations for each Topic, problems generally progress from simpler to more complex, incorporating knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, in Algebra I, creating linear models for data in Topic 6 incorporates and builds on rate of change from Topic 4.
• As students progress through the Overview, Exploring, and Summary sections, the Practice (Guided and More), Assessment (Automatically Scored and Constructed Response), and Activity Sheets sections are placed intentionally in the sequencing of the materials to help students build their knowledge and understanding of the mathematical concepts addressed in the Topic.
• The MARS Tasks are also placed intentionally in the sequencing of the materials to support the development of the students’ knowledge and understanding of the mathematical concepts that are addressed by the tasks.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having a variety in what students are asked to produce.

Throughout a Topic, students are asked to produce answers and solutions as well as explain their work, justify their reasoning, and use appropriate models. The Practice section and Automatically Scored items include questions in the following formats: fill-in-the-blank, multiple choice with a single correct answer, and multiple choice with more than one correct answer. Constructed Response items include a variety of ways in which students might respond, i.e. multiple representations of a situation, modeling, or explanation of a process. Also, the types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced, but as students progress in their knowledge, students are expected to transition to more efficient solution strategies or representations.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. The materials include a variety of virtual manipulatives, as well as, integrate hands-on activities that allow the use of physical manipulatives.

Most of the physical manipulatives used in Agile Mind are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students also have the opportunity to represent proportional relationships virtually with a table and graph and generate random samples to draw inferences. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, in Geometry Topic 23, Relating 2-D and 3-D objects, students use models of prisms, cones, and spheres that can be cut. In addition, they use modeling clay, fishing wire (or dental floss), and linking cubes throughout the topic.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. The Deliver Instruction section for each Block of a Topic includes Framing Questions for the start of each lesson. For example, in Algebra 1, Topic 8 Block 6, the Framing Questions are: “Would you use an equation or an inequality to describe this situation? How many variables will you need to describe this situation? Why?” During the lesson, the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the the end of each lesson, Deliver Instruction includes Further Questions. For example, in Geometry, Topic 7 Block 3 “Why can a triangle never have two obtuse angles? Two right angles? How could knowing the sum of the angles of a triangle help you find the sum of the angles of a quadrilateral? What about any polygon?”

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for containing 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, the materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials contain Professional Support which includes a Plan the Course section and a Scope and Sequence document. The Plan the Course section includes Suggested Lesson-planning Strategies and Planning Resources. Each Topic contains an Advice for Instruction section, and that is divided into Prepare Instruction and Deliver Instruction. For each Topic, Prepare Instruction includes Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources, and Language Support, and for each Block within a Topic, Deliver Instruction includes Agile Mind Materials, Opening the Lesson, Framing Questions, Lesson Activities, and Suggested Assignment. In Lesson Activities, teachers are given ample annotations and suggestions as to what parts of the materials should be used when and Classroom Strategies that include questions to ask, connections to mathematical practices, or statements that suggest when to introduce certain mathematical terms or concepts.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. For example, in Algebra II, Topic 8 Block 5 teachers are directed to, “Use the animation on page 1 to introduce the idea of area varying with more than one dimension. As you view each new panel, have students respond to the appropriate questions on their Student Activity Sheets. Then, play the panel to confirm their responses. [SAS 4, questions 1-5]”

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for containing 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.

In Professional Support, Professional Learning, there is a group of four interactive essays in each course entitled “Developing Concepts Across Grades”, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Algebra II, Geometry and beyond. These interactive essays give teachers the opportunity to not only make connections between the courses they are teaching and previous courses, but they also give teachers the opportunity to improve their own knowledge in regards to connections that will be made between the courses they are teaching and future courses.

In addition to “Developing Concepts Across Grades”, each course also contains a section of interactive essays entitled “Going Beyond (course name)”. In Algebra I, there are three essays in this section: Average and Instantaneous Rates of Change, The Slope of a Curve, and The Relationship Between Exponential and Logarithmic Functions. In Algebra II, there are two essays in this section: Linearizing Data Using Logarithms, and From Rates of Change to Derivatives. In Geometry, there are three essays in this section: Trigonometric Functions, Understanding Area of Irregular Shapes using Calculus, and Radians. Along with having their own section in Professional learning, each of these essays are also referenced in Deliver Instruction for the Blocks where they are appropriate under the title of Teacher Corner. For example, in Geometry, the essay Trigonometric Functions is referenced for teachers in Block 2 of Topic 15, Right Triangle and Trig Relationships, or in Algebra II, Linearizing Data using Logarithms is referenced in Block 3 of Topic 14, Logarithmic Functions.

In Professional learning, there are also sets of Video or Print essays. The Print essays are divided as either Curriculum or Course Management Topics, and there is a series of three essays in Algebra II titled “Rational Functions and Crossing Asymptotes” that addresses mathematical concepts that extend beyond the current course. The Video Essays are: Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction.

The instructional materials reviewed for Agile Mind Traditional series meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels. The materials provide assessments that are specifically designed for the purpose of gathering information about students’ prior knowledge, and the materials also include indirect ways for teachers to gather information about students’ prior knowledge if teachers decide to use them that way.

Each course includes additional Topics intended to assess students’ prior knowledge. Algebra I includes computations with rational numbers and foundations of solving equations. Geometry includes computations with rational numbers and foundations of functions and linear equations. Algebra II includes computations with rational and irrational numbers, operations with exponents, and foundations of linear and quadratic functions and equations.

In Prepare Instruction for each Topic, there is a set of Prerequisite Skills needed for the Topic, and the Overview for each Topic provides teachers with an opportunity to informally assess students prior knowledge of the Prerequisite Skills. For example:

• In Algebra I, Topic 1, Advice for Instruction, the prerequisite skills required for the lesson are: “Reading and constructing graphs, Domain and range and Exponential and quadratic patterns in data”.
• In Algebra I, Topic 9, About this Topic has several references to framing students’ thinking, “This topic, Absolute value and other piecewise functions, builds on students' understanding of the absolute value of a number and of the absolute value of a difference of two numbers as a distance on the number line to develop the absolute value function..”
• In Algebra II, Topic 1, Deliver Instruction, Overview states, “Classroom strategy, The material on these two pages are designed to activate students' prior learning from previous courses, but keep it in the context of setting the stage for new learning in this course. Do not succumb to the temptation of re-teaching everything students should have learned in prior courses. Instead, use the material on these pages to actively engage students in recall of prior work, facilitating students' conversations to resurface what they have learned previously about these key function families. This will set students up for success not only for this topic but also for work in future topics with new function families.”

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. Across the series, common student errors and misconceptions are identified and addressed in Deliver Instruction as parts of “Classroom Strategy”, but “Classroom Strategy” is not solely used for identifying and addressing common student errors and misconceptions.

• In Algebra I Topic 6, Deliver Instruction for Block 3 states, “Another common mistake is for students to look at the differences in the y-values only, and not relate these changes to the differences in the x-values. This mistake sometimes comes from misconceptions students create when exploring linear data where the x-values only increase by 1 unit. Throughout all data interpretation in tables, refer to the ratio or rate of change. If the students say that y-values are increasing by some number, ask them to complete their sentence by adding a description of how the corresponding x-values are changing, even when the change in x is only 1 unit.”
• In Geometry Topic 22, Deliver Instruction for Block 2 states, “Students often mistakenly use diameter in computations instead of radius and vice versa. Similarly, students often lose track of when to use a 1/2 or 2 in computations. This can be especially confusing when looking at something like half of a circumference.”
• In Algebra II Topic 11, Deliver Instruction for Block 4 states, “Be sure to discuss with students the importance of isolating the radical. Show students an example of what happens if they don’t. Also remind them about how to expand a binomial. Be sure they do not distribute the exponent over addition or subtraction.”

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide opportunities for ongoing review and practice, and feedback occurs in various forms. Within interactive animations, students submit answers to questions or problems and feedback is provided by the materials. Practice problems and Automatically Scored Assessment items are submitted by the students, and immediate feedback is provided letting students know whether or not they are correct and, if incorrect, suggestions are given as to how the answer can be improved. The Lesson Activities in Deliver Instruction provide some suggestions for feedback that teachers can give while students are completing the lessons.

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Each Topic consists of three main sections, Overview, Exploring, and Summary, and these three sections are divided into Blocks. Each Block contains lesson activities, materials for Practice, Assessment, and Activity Sheets, along with a MARS Tasks if applicable for the Topic. In each Topic, the Blocks and lesson activities are sequenced for the teacher. In the Advice for Instruction for each Topic, Deliver Instruction for each Block contains instructional notes and classroom strategies that provide teachers with key math concepts to develop, sample questions to ask, ways in which to share student answers, and other similar instructional supports.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation that materials embed tasks with multiple entry­-points that can be solved using a variety of solution strategies or representations. Overall, tasks that meet the expectations for this indicator are found in some of the Constructed Response Assessment items and Student Activity Sheets that are a part of all Topics. MARS Tasks embedded in some of the Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, Geometry, Topic 14 Mars Task: Garden Chair, students determine an angle made by the wooden construction of the chair. Students can begin the problem by using either the angle sum theorem or the exterior angle theorem. Students could find all the angles in the problem first or the minimum required for the problem.

The instructional materials reviewed for Agile Mind Traditional series meet the expectation that the materials suggest accommodations and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The materials provide suggestions for English Language Learners and other special populations in regards to vocabulary and instructional practices throughout each course in the series. In Prepare Instruction for Topic 1 of each course, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, and that essay describes general strategies that are used across the series such as a vocabulary notebook, word walls, and concept maps. Teaching Special Populations of Students also describes general strategies that are used across the series for other special populations, and these strategies include progressing from concrete stage to representational stage to abstract stage and explicitly teaching metacognitive strategies through think alouds, graphic organizers, and other visual representations of concepts and problems.

In addition to the general strategies mentioned in Teaching Special Populations of Students, there are also many specific strategies listed across each course of the series in Deliver Instruction. In Deliver Instruction, Support for ELL/other special populations includes strategies that can be used with both English Language Learners and students from other special populations, and strategies specific to other special populations can also be found in Classroom strategy or Language strategy. An example of Support for ELL/other special populations from Geometry, Topic 1, Block 5, Pages 2-3 is “This puzzle acts much like a Cloze activity, in which key vocabulary words are removed from a paragraph, to build confidence and quickly assess fluency with the vocabulary. This type of activity can be particularly helpful to reinforce key understandings for students with a variety of learning differences, including challenges with language acquisition and processing. ELL students should add the labeled diagram to their vocabulary notebooks.” An example of a strategy for other special populations from Algebra II, Topic 13, Block 4, Page 10 is “Language strategy. You may wish to use a paired reading strategy for this page. In paired reading, one student in the pair reads the first sentence to the other student in the pair. The second student then paraphrases what was read back to the first student. Then, the students switch roles and repeat the process for the next sentence. This continues until the entire page is read and processed. This strategy can be modified if one student in the pair has a reading challenge so that only one student reads the passage, but both students take turns paraphrasing what was read.”