High School - Gateway 3

The instructional materials reviewed for Agile Mind Integrated 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 include MARS Tasks where students apply concepts they have learned within the topic and connect concepts from other topics.

The instructional materials reviewed for Agile Mind Integrated 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 Mathematics I, Topic 3,  Functions, the Exploring section consists of Function Notation, Modeling with Functions and Graphs.  

Within the explorations for each topic, problems progress from simpler to more complex, incorporating knowledge from prior problems or topics. This progression helps students to make connections among mathematical concepts.  For example, in Mathematics I, Topic 5, Moving Beyond Slope Intercept, students create linear models for data that incorporates and builds on content from Topic 4, Rate of Change.

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 Integrated series meet the expectations 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. The Student Activity Sheets provide opportunities for students to explain their thinking and show their own approach to solving a problem. 

The types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced.  As students progress in their knowledge, they transition to more efficient solution strategies and representations.

The instructional materials reviewed for Agile Mind Integrated 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 hands-on activities that allow the use of physical manipulatives as well as a variety of virtual manipulatives.

Most of the physical manipulatives used in Agile Mind Integrated are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students use virtual manipulatives to develop a conceptual understanding of the mathematics in that topic. Each topic has a Prepare Instruction section that lists the materials needed for the topic. Manipulatives accurately represent the related mathematics. For example, in Mathematics II, Topic 3, Operations on Polynomials, Exploring, Multiplying Polynomials, students use virtual algebra tiles to multiply binomials.  This relates back to the area model used in multiplying multi-digit numbers.

The instructional materials reviewed for Agile Mind Integrated 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 Mathematics I, Topic 7, Descriptive Statistics, Block 6, students “think about the different measures of center and spread they have learned about in this topic.”  Students consider “When is it more appropriate to report the mean? Median? When is it more appropriate to report the standard deviation? Interquartile range?”
• In Mathematics I, Topic 7, the students consider, “In which ways can you compare two different data sets? How do you know which statistic to use when comparing data?”
• In Mathematics II, Topic 1, Absolute Value and Other Piecewise Functions, Block 2, students access their prior knowledge such as,  “In the functions you have previously studied, how did you represent a vertical translation of the function? How did you represent a horizontal translation? A vertical stretch or compression?”   

At the end of each lesson, Deliver Instruction included Further Questions that support a deeper exploration of the mathematics content.  For example:

• In Mathematics II, Topic 1, Absolute Value and Other Piecewise Functions, after students have explored the concept using an interactive app, they consider “How will changing the value of a to 3 affect the graph? How will changing the value of h to 2 affect the graph?”
• In Mathematics II, Topic 8, Rational Expressions and Functions, students consider “Do all rational functions have vertical asymptotes? If no, give an example of one that doesn't. Can the graph of a function ever cross a horizontal asymptote? Explain.”

The instructional materials reviewed for Agile Mind Integrated 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 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 when each part of the materials should be used. The Classroom Strategies include questions to ask, connections to mathematical practices, and 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, the following exchange occurs in the Deliver Instruction section of Mathematics II, Topic 2, Introduction to Quadratic Functions, Block 1:

• The teacher “play[s] the animation on page 1 to demonstrate the tiling project, pausing after a 2' × 2' area is laid with the tiles (4 total tiles).”  
• Students “follow along by coloring in appropriate areas on their own square grids and fill in the data values in their tables for the first two edge lengths, and then to complete their diagrams and tables for edge lengths of 3, 4, and 5.”  
• The Teacher “play[s] the animation on page 2 to highlight the patterns in the data.”
• Students “develop a rule to generalize the pattern in the data.” 
• Students “use their graphing calculators to make a scatterplot of the data from the table and then to graph the function rule over the scatterplot.”

The instructional materials reviewed for Agile Mind Integrated 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 Mathematics I, Mathematics II, Mathematics III, 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 This Course.” In Mathematics 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 Mathematics II, there are two essays in this section: "Trigonometric Functions" and "Radians." In Mathematics III, there are three essays in this section: "Linearizing Data Using Logarithms," "From Rates of Change to Derivatives," and "Understanding Area of Irregular Shapes using Calculus." 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.

In Professional Learning, there are also sets of Video or Print Essays. The Print Essays are divided as either Research to Practice or Content to Pedagogy, which, in Mathematics III, includes a series of three essays titled “Rational Functions and Crossing Asymptotes” that addresses mathematical concepts that extend beyond Mathematics III. 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 Integrated 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. Mathematics I includes computations with rational numbers and foundations of solving equations. Mathematics II includes computations with rational numbers and foundations of functions and linear equations. Mathematics III 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 Mathematics I, Topic 1: Constructing Graphs, Advice for Instruction, the prerequisite skills are “Plotting points and labeling axes, Identifying independent and dependent variables, Reading data from a table or from a graph.”
• Int Mathematics II, Topic 1: Absolute value and other piecewise functions, the heading “About This Topic” has several references to framing students’ thinking, “This topic, Absolute value and other piecewise functions, builds on students' prior work with absolute value functions in earlier courses and deepens their knowledge of this function type while also introducing students to other piecewise functions.”
• In the Deliver Instruction section for Mathematics III, Topic 1: Arithmetic and geometric sequences and series, guides the teacher, “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 Integrated 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. Examples of supports from the materials include:

• In the Deliver Instruction section for Mathematics I, Topic 9: Absolute Value Functions and Equations, guidance for absolute values is given: “Classroom strategy. Some students may incorrectly think that absolute value means to take the opposite of any number; others may think that the solutions to an absolute value equation are always a number and its opposite. Encourage students to use the values in the table they built as they think through the statements in the puzzle. Students often think that a variable with no sign or a plus sign is positive, and one with a minus sign is negative. You can address this misunderstanding by relating the numbers in the table to the variable in the puzzle. Explain that the variable can stand in for a positive or a negative number, even though the sign is not written.”
• In the Deliver Instruction section for Mathematics II, Topic 4: Solving Quadratic Equations, guidance for similar graphs is given: “Classroom strategy. When modeling motion data with quadratic functions (as in the case of the balloon launcher) students will often confuse the actual path (which is height versus horizontal distance) with the graph of height versus time. Both are parabolic, so spend some time relating the two. It may be helpful to spend some time looking at the path of the balloon in the Overview with the associated table of distances and times and relating the times to the graph of the path through the common distances.”
• In the Deliver Instruction section for Mathematics III, Topic 1: Arithmetic and geometric sequences and series, guidance for connecting numeric and geometric ideas is given: “Classroom strategy. Give students equilateral triangle pattern blocks to help them make sense of the problem. This will support their understanding of how the perimeter grows and help them discern the pattern and underlying structure in the problem, connecting the numerical representation of the perimeter to the geometry of the figures. This strategy is particularly important for some students with learning differences to help them move from concrete to abstract representations.”

The instructional materials reviewed for Agile Mind Integrated 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 Integrated series 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 Overview introduces the mathematics concept with a real-world example. The Deliver Instruction provides guiding questions that asks students questions about the real-world scenario. The scaffolding within the Overview and Exploring provide a pace that allows students to focus on a particular idea or concept in small pieces. In the Advice for Instruction for each topic, Deliver Instruction for each block contains instructional notes and classroom strategies that provide teachers with key mathematics 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 Integrated 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, in Mathematics II, Topic 10: Lines, Transversals, and Triangles, Exploring, A Triangle or Not?, pages 2-6, students can explore the difference of rigidity between a triangle and a square in various ways, including using straws and string or an online interactive app.

The instructional materials reviewed for Agile Mind Integrated 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. Examples include:

• The Deliver Instruction for Mathematics III, Topic 8: Rational Expressions and Equations, Block 1 provided the guidance, “Support for ELL/other special populations. Provide the sentence stem ‘A rational function is _______. An example of a rational function is _______.’ Consider creating a concept map to help students. Make sure the concept map of rational function includes a definition, examples and non-examples. Sentence stems like this one help students with some learning differences focus their thinking.”
• In Mathematics I, Topic 4: Rate of Change, Block 4, Language Strategy, teachers are directed “Have students practice using the language of direct variation as it applies to the blue jeans situation. For example, asking students to restate the statement ‘y is proportional to x’ in terms of the blue jeans context will help the terms make more sense to them. Choral responses like these help students with some learning differences, such as auditory processing disorders, appropriately use and internalize the meaning of academic language.”
• In Mathematics II, Topic 4: Solving Quadratic Equations, Block 1 provides a strategy to support ELLs/other special populations with graphs of piecewise functions, “Ask students to sketch 2-3 other graphs that they think represent piecewise defined functions. After students have sketched their graphs, have them turn to a partner and explain why they think their examples represent piecewise defined functions. A sentence frame such as ‘This function is a piecewise function because one piece can be modeled with the function _______ and another piece can be modeled with the function _______.’ may help students understand that piecewise defined functions are composed of two or more different functions. Be on the lookout for students who create piecewise functions that are composed of three or more linear functions or that use non-linear functions. Monitor the conversations as students are talking and choose some students to share their graphs and explanations with the whole class.”