6th Grade - Gateway 3

The instructional materials reviewed for Agile Mind Grade 6 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 where 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 and give 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 items. 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 Grade 6 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.
• Within the explorations for each Topic, problems generally develop from more simple to more complex problems and incorporate knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, using equations and inequalities to solve problems in Topic 11 builds upon writing algebraic expressions, equations, and inequalities from Topics 9 and 10.
• 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 Grade 6 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 Grade 6 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 equations or functions virtually with tables and graphs. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, Topic 13, Surface Area and Volume, students work with physical unit cubes and a cardboard box, along with having virtual representations of them, in order to expand their understanding of volume to include rectangular prisms whose edge lengths are not whole numbers.

The instructional materials reviewed for Grade 6 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 Topic 8 Block 4 the Framing Question is: “Have you ever used a map that had a grid system for naming points?” During the lesson the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the end of each lesson, Deliver Instruction includes Further Questions. For example, in Topic 8 Block 4 “If both coordinates are positive, what does that tell you about the location of a point? If both coordinates are negative, what does that tell you about the location of a point? If one coordinate is positive and the other is negative, what does that tell you about the location of the point?”

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for containing a teachers edition 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, 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 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 Topic 7 Block 2 teachers are directed to, “Use the interactive animation to explore a distance vs. time graph. Use these questions to guide student discussions during the activity: When you drag Terrence away from the motion detector, what does the graph look like? Why? [When you drag him away, his distance from the motion detector increases. Since the graph measures Terrence's distance from the motion detector at any given time, this causes the graph to rise, or get farther away from the x-axis, as you move from left to right on the graph.]; How does dragging him back toward the motion detector affect the graph? Why? [When you drag him toward the motion detector, his distance from the motion detector decreases. Since the graph measures Terrence's distance from the motion detector at any given time, this causes the graph to fall, or get closer to the x-axis, as you move from left to right on the graph.]; What does the graph look like when you drag him at a fairly slow, steady rate away from the motion detector? [The graph will be a line that rises from left to right, but will not be very steep.] What if you drag him away from the motion detector faster, but still at a steady rate? [The graph will still be a line that rises from left to right, but it will be steeper than the first line.]; and When Terrence stops during his skate, what does the graph do? Why? [The graph will be a horizontal line. This happens because when Terrence is not moving, his distance from the motion detector is not changing, but time continues to pass.]”

The instructional materials reviewed for Agile Mind Grade 6 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 are four interactive essays entitled “Developing concepts across grades”. There is an Overview that explains the purpose of the four essays, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay is accessible to teachers and not students, and the Overview states “these essays are available for educators to illustrate connections and deepen understanding around what students may have already learned, and where they are headed on their journey.” Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Geometry, Algebra II, and beyond. By examining the progressions of the concepts beyond Algebra II, teachers have the opportunity to improve their own knowledge of more advanced mathematics concepts that build upon grade-level standards. For example, in Volume, teachers progress from packing a right, rectangular prism with unit cubes to developing the formulas for the volume of cylinders and cones to finding the volume of a figure generated by rotating a two-dimensional shape around a horizontal axis. Also, in Proportionality, teachers explore how proportional relationships are part of the following mathematical concepts: scaling images, linear functions, trigonometric ratios, rational functions, and the derivative.

In addition to “Developing concepts across grades”, the Grade 6 materials also contain a section of interactive essays entitled “Going beyond Grade 6”. There is an Overview that explains the purpose of the three essays, and the Overview states, “These essays are not intended for use with your 6th grade students; rather, they are designed to provide you with important connections and background that will support you as you help your students master 6th grade content.” The topics for these three essays are Rate of Change, Understanding Area of Irregular Shapes, and Calculating Outliers. 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 Topic 3, the essay Rate of Change is referenced for teachers in Block 1 as part of the Overview, and in Topic 12, Understanding Area of Irregular Shapes is referenced in Block 5 on page 4. The three essays connect the Grade 6 content to advanced mathematical concepts through multiple grades and courses.

In Professional Support, there is a section of Professional Essays which are in either Print or Video format. The Print essays are divided as either Curriculum or Course Management Topics, and although some of the Curriculum Essays are content specific, they do not address mathematical concepts that extend beyond the current grade. The Video Essays - categorized into Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction - do not directly provide adult-level explanations or examples of advanced mathematics concepts.

The instructional materials reviewed for Agile Mind Grade 6 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. 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 Grade 6 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 including Practice, Assessment, and Activity Sheets, along with any MARS Tasks in the Topic. In each Topic, the Blocks are sequenced for the teachers, and the lesson activities within the Blocks are sequenced for the teachers. 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 Grade 6 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 Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, in Topic 8 Constructed Response 1 students are instructed to "Describe four more possible events in the life of Andre or his family. For each event describe when it happened. Then graph and label the rational number representing that event. Use at least one negative rational number, at least one whole number, and at least two fractions or decimals." Another example is in Topic 12 Constructed Response 2. Students are asked to find the area and/or perimeter of different figures that are superimposed on a grid. Students are given the scale for the grid.

The instructional materials reviewed for Agile Mind Grade 6 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. In Prepare Instruction for Topic 1, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, which 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, including 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. 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 Topic 1, Block 4, Page 5 is “Provide students who have handwriting challenges or other language challenges with index cards that contain the 4 rules of the Order of Operations. Having the text already written will allow these students to focus on understanding the meaning of each rule instead of transcribing the words. While the rest of the class is writing the rules, encourage these students to write an example next to each rule to help them process the meaning. Have them circle each rule in a different color. Then have them write the expression from the animation and color code each step in the expression with the color of the appropriate rule.” An example of a strategy for other special populations from Topic 8, Block 1, Page 1 of Exploring Positive and Negative Rational Numbers is “Classroom strategy. Bank balances will use negative numbers, but when we talk about those balances using the word “debt” we use the absolute value. We say, “I have a debt of $15,” rather than, “I have a debt of $-15.” Or, stated another way, “I have a balance of $-15, so my debt is $15.” This is a subtle distinction, especially for students at this age, and complete mastery of this concept is not expected on this page. Throughout the topic, as students are discussing debt, continue to reinforce the notion that the greatest debt is the balance with the least value, or the negative number furthest from zero. Be aware that some students with learning differences may continue to need more explicit instructions and repetition.”

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics. The activities are diverse, meeting the interests of a demographically, diverse student population. The names, contexts, videos, and images presented display a balanced portrayal of various demographic and personal characteristics.