Report for High School
Overall Summary
The instructional materials reviewed for the Agile Mind Traditional series meet expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials attend to the full intent of the non-plus standards and allow students to fully learn each non-plus standard, but they do not attend to the full intent of the modeling process when applied to the modeling standards. The materials regularly use age-appropriate contexts, apply key takeaways from Grades 6-8, and vary the types of numbers being used, but the materials do not explicitly identify and build on knowledge from Grades 6-8, although they do foster coherence through meaningful connections in a single course and throughout the series. In Gateway 2, the instructional materials meet the expectation that materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials reviewed meet the expectations for the development of overarching, mathematical practices; reasoning and explaining; modeling and using tools; and seeing structure and generalizing.
High School
Alignment
Usability
Overview of Gateway 1
Focus & Coherence
Criterion 1.1: Focus & Coherence
The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. Overall, the instructional materials attend to the full intent of the non-plus standards and allow students to fully learn each non-plus standard, but they do not attend to the full intent of the modeling process when applied to the modeling standards. The materials regularly use age-appropriate contexts, apply key takeaways from Grades 6-8, and vary the types of numbers being used. The materials do not explicitly identify and build on knowledge from Grades 6-8, although they do foster coherence through meaningful connections in a single course and throughout the series. The instructional materials spend a majority of time on the widely applicable prerequisites from the CCSSM.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Although there are a few instances where all of the aspects of the standards are not addressed, most non-plus standards are addressed to the full intent of the mathematical content by the instructional materials.
The following are examples of standards that are fully addressed:
- A-APR.3: Algebra I Topic 18 Roots, Factors, and Zeros connects x-intercepts to zeros to factors. In Algebra II Topic 5 Polynomial Functions there are two lessons, Long Term Behavior and Zeros and Higher Degree Polynomials, where students factor polynomials to find zeros and use zeros to construct polynomial functions. Also in Algebra II Topic 6 Polynomial equations - Theorems of Algebra, students use Theorems of Algebra (such as The Fundamental Theorem of Algebra and Remainder Theorem), and factorizations to find zeros in order to graph the polynomial function.
- F-BF.4a: In Algebra II Topic 2 Understanding Inverse Relationships students find equations of inverses of linear, exponential, and quadratic functions and give restrictions where needed.
- G-SRT.8: In Topic 14 Pythagorean Theorem and the Distance Formula and Topic 15 Right Triangle and Trig Relationships of Geometry, students use the Pythagorean Theorem as well as the trigonometric ratios to solve right triangles. In addition to the lesson demonstrations, student activity sheets, practice, and assessment items, both topics include MARS tasks which fully address the intent of this standard by providing students opportunities to solve right triangles using the trigonometric ratios and Pythagorean Theorem in applied problems.
- S-ID.2: In Algebra I Topic 7 Descriptive Statistics students compare data sets using mean and median in the lesson Measures of Center. In the next lesson, Measures of Spread, students compare data sets using range and standard deviation.
The following standard is partially addressed:
- G-CO.13: While no instruction was provided on G-CO.13, there is one instance where this standard is assessed, in Topic 15 of Geometry Constructed Response Assessment #1. Students inscribe an equilateral triangle in a circle, but students are not provided an opportunity to practice this concept in the lesson materials. Constructions can be found in Geometry Topic 11 Compass and Straightedge Constructions; Geometry Topic 19 Chords, Arcs, and Inscribed Angles; and Geometry Topic 20 Lines and Segments on Circles; however, students are not given an opportunity to construct a shape inside of a circle.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. Overall, most of the modeling standards are addressed with various aspects of the modeling process present in isolation or combination. However, opportunities for the full modeling process are absent.
The materials often allow students to incorporate their own solution method to find a predetermined quantity. Modeling opportunities in the materials are thus “closed” in the beginning and the end while “open” in the middle. In many instances, materials step students through the modeling process using a series of questions and/or prompts. In addition, students are rarely given the opportunity to question their reasoning and “cycle” through the modeling process by validating their conclusions and potentially making improvements to their model.
The following examples allow students to engage in only a part of the modeling process:
- In Algebra I Topic 18 Student Activity Sheet 2 (A-SSE.3a), students are given a function rule of a real-world context and guided through a series of questions, mostly directed by the teacher with questions and/or Exploring “Solving by graphing.” The same is true on Student Activity Sheet 4, which follows along with Exploring “Roots, factors, and zeros.” Students are given a real-world context and taken through a series of questions, as posed on the Student Activity Sheet. Students use a variety of tools to find a solution to a quadratic equation. Students do not define their own variables or formulate the equation or function needed to work the problem. The materials provide students with a graph with predefined axes and scale as well as the function they are to graph.
- In Algebra I Topic 14 (A-SSE.3c, F-BF.1a), there are two sections in the Exploring part of the Topic that starts out with the word “Modeling.” Students follow step-by-step directions on how to apply the modeling context (make a table, graph the data, answer questions, etc.). The Constructed Response assessment item for the topic gives students exact measurements when starting the problem, steps them through by telling them which tools to use, and does not have them justify their solution; therefore, students do not complete the modeling cycle. Student Activity Sheets 2-5 have application problems where students are asked to do things such as explain, describe, and discuss; students are also asked to check the validity of their answer. However, there isn’t an opportunity for students to complete the full modeling process in any one problem.
- In Algebra I Topic 8 (A-CED.2) Student Activity Sheet 3 Question 26 is an example of an application problem where students choose their tools to use in order to solve the problem. However, there are exact values given to students leading to one correct answer. In addition, students are not required to provide any justification for or validate their solution. The same thing happens in Student Activity Sheet Question 31.
- In Algebra 1 Topic 10 (A-CED.3) there is a MARS task which asks students to explain their work. However, all quantities are fixed, and students are not asked to check the validity of their solution or to adjust as necessary. There is a Constructed Response question on the Assessment that is an application where students are required to identify the variables, write a system, use a graph or table to solve, and then show how to check the answer. They are taken step-by-step through the process.
- In Algebra II Topic 11 (F-IF.5) a Constructed Response Assessment question has students find the domain and range, in context, and relate it to the context of the situation. Students are also asked to justify their answer in another part of the problem; however, students do not develop the model.
- In Geometry Topic 15 (G-SRT.8) Student Activity Sheet 3 Question 19 is also found in Exploring Right triangle and trig relationships. Students are presented with an open-ended question but given specific variables to use in order to solve it. Students are not given an opportunity to define the variables. In addition, students are not asked to validate or interpret their solutions. All application problems in this Topic are routine and require one or two steps. An example of this can be found in Student Activity Sheet 3 Question 21 where students are given the context and a labeled picture to find the solution to the problem which asks students to find the height of the cliff. Students are required to use trigonometric ratios to solve for the height given the angle of elevation and the horizontal distance from the cliff to the boat. A second example is found in Assessment for this Topic; Constructed Response 3 has students solve a few problems using the context of a lighthouse used to orient ships.
- In Geometry Topic 14 (G-GPE.7) Question 12 on Student Activity Sheet 2 gives measurements and asks students if the door frame is rectangular. Students are directed to justify their response. They are not afforded the opportunity to complete the modeling process. Student Activity Sheet 3 requires students to support their response but requires that support be in a diagram.
- In Algebra I Topic 7 (S-ID.1, S-ID.2, and S-ID.3) on Student Activity Sheet 2 Question 12 students are asked to predict, explain, and check their prediction by calculating the mean and median. Students are given the tools to use throughout the step-by-step questions. Question 13 and 14 are contextual problems where students create histograms. Students explain their reasoning if something was changed based on the histograms. Students use a context but are not able to formulate the variables. The data set to be used is given at the beginning, and students are given questions to guide them through. In Student Activity Sheet 4 Question 17 students create a survey for the class, conduct it, and do specific things to interpret the survey. However, students are not asked to verify their responses.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectations for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers (WAPs). (Those standards that were not fully attended to by the materials, as noted in indicator 1ai, are not mentioned here.)
In the Algebra I course, students spend most of their time working with WAPs from the Algebra, Functions, and Statistics and Probability categories. During the Geometry Course, students spend most of their time working with WAPs from the Geometry category. The Algebra II course focuses on the WAPs in the Functions, Algebra, and Geometry categories. Within the Algebra I and Algebra II courses, students also spend time on the Number and Quantity WAPs.
Examples of students engaging with the WAPs include:
- Algebra I: In Topic 13 Law of Exponents students are provided with multiple opportunities to explore and interpret laws of exponents using scenarios such as fuel consumption and distance from the sun to the Milky Way Galactic Center (N-RN.1,2). Topic 13 covers the general rules for exponents as well as scientific notation. Topics 16 and 18, Operations on Polynomials and Solving Quadratic Equations, provide several opportunities to explore the structure of an expression to identify ways to rewrite it and to factor a quadratic expression to reveal the zeros of the function it defines. The topics provide practical illustrations using blueprints from a construction site to illustrate finding sums and differences of two polynomials and a water balloon launch to illustrate solving quadratic equations (A-SSE.2,3a).
- Geometry: Topics 9, 10, 12, and 13 address similarity and congruence as referred to in G-SRT.5. Topics 9 and 10 focus on congruence, and Topics 12 and 13 focus on similarity.
- Geometry: In Topics 4, 5, and 6 students prove theorems about lines and angles (G-CO.9). Proofs begin in Topic 4 on Student Activity Sheet 2 with algebraic proofs. The topic then progresses in Student Activity Sheet 3 as materials provide multiple proofs for students to “fill in the blank” for the missing part. In Student Activity Sheet 4 there are multiple cases where students are expected to complete the majority of an entire proof. In Topic 5 proofs continue in Student Activity Sheet 1 as well as indirect proofs in Student Activity Sheet 4.
- Algebra II: In Topic 1 Student Activity Sheet 3 students work with geometric series in word problems and are asked to write a function rule that models the given situation (A-SSE.4). Throughout Student Activity Sheet 3 students are exposed to finite, geometric series by using the general formula and finding sums.
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The instructional materials reviewed for the Agile Mind Traditional series, when used as designed, meet the expectation for allowing students to fully learn each non-plus standard. Overall, there are multiple opportunities for students to fully learn the non-plus standards by engaging with all aspects of the standards and not distracting students with prerequisite or additional topics. Examples of the standards where students have multiple opportunities to fully learn the standard include, but are not limited to:
- A-SSE.1a: Algebra I Topic 2 Exploring Tiling Square Pools offers students the opportunity to interpret parts of expressions as they examine different representations for determining the number of tiles needed in a pattern to create a border around a pool. Constructed Response 1c of Topic 2 offers students the opportunity to interpret parts of an expression as they create a symbolic representation of the relationship between the length of the side of a square flower bed and the perimeter of the flower bed. Throughout the remainder of Algebra I and into Algebra II, there are multiple opportunities for students to interpret parts of an expression, and some of those opportunities are:
- Algebra I Topic 6 More Practice Problem 23 has students interpret parts of an equation in order to determine which conclusion can be made based on the equation and its accompanying graph.
- Algebra I Topic 14 includes multiple opportunities for students to interpret parts of expressions, equations, and functions in different contexts that represent exponential growth and decay.
- Algebra II Topic 1 Exploring Arithmetic Sequences and Series has opportunities for students to interpret parts of expressions that represent the same arithmetic sequence, and Algebra II Topic 1 Exploring Geometric Sequences and Series has students interpret parts of expressions while comparing different expressions that represent the same geometric sequence.
- Algebra II Topic 13 Guided Practice Problems 11 and 12 have students interpret parts of a general exponential equation in order to determine how to substitute numerical values into the equation, and More Practice Problems 3 and 6 has students selecting which exponential equation models a situation which means the students interpret parts of the exponential expression to choose the correct equation.
- Algebra II Topic 21 Automatically Scored Problem 10 has students interpret parts of a trigonometric expression in order to choose which trigonometric equation best represents a given situation.
- A-APR.6: In Algebra 1 Topic 16 students are introduced to dividing polynomials, and the problems include dividing by monomials with remainders. In Algebra II Topic 6 Exploring Theorems of Algebra students divide polynomials by linear binomials as they engage with The Remainder Theorem, and in Algebra II Topic 9 Exploring Rational Expressions, students use polynomials to build rational expressions by dividing polynomials using factoring techniques with no remainders. In the remainder of Algebra II Topic 9, students further develop their skills in rewriting simple rational expressions as they use long division with expressions that involve remainders in order to analyze the graphs of rational functions that correspond to the rational expressions.
- G-GPE.5: In Algebra 1 Topic 5 Student Activity Sheet 3 Problems 18 and 19 students informally use the slope criteria for parallel and perpendicular lines to solve geometric problems by writing equations of lines that are parallel and perpendicular to given lines, and they do the same thing in More Practice Problem 16 of the same topic. In Geometry Topic 6 Exploring Lines and Algebra, students formally derive the slope criteria for parallel and perpendicular lines. Student Activity Sheet 3 of the same topic, along with More Practice Problems 17 and 19 and Automatically Scored Problems 12 and 13, gives students opportunities to write the equations of lines parallel and perpendicular to given lines. Geometry Topic 8 Constructed Response Problems 2 and 3 have students use the slope criteria for parallel and perpendicular lines to solve geometric problems by having the students find the centroid, orthocenter, and Euler line for triangles with given coordinates.
There are non-plus standards where the materials provide students an opportunity to fully learn the standard, and the materials could solidify the students’ learning with more opportunities that address the standard:
- F-IF.8b: In Algebra 1 Topic 14 and Algebra II Topic 13 there are problems where students interpret expressions in exponential functions, and there could be more opportunities for students to use properties of exponents to interpret functions.
- G-SRT.7: In Geometry Topic 15 students work with trigonometric ratios, and throughout the topic students engage with problems involving complementary angles. Students’ understanding of G-SRT.7 could be further solidified by offering more opportunities for students to use the relationship between the sine and cosine of complementary angles.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts, apply key takeaways from Grades 6-8, and vary the types of real numbers being used.
The materials use age appropriate and relevant contexts throughout the series. The following examples illustrate appropriate contexts for high school students.
- In Algebra I, Topic 14 gives the growth of a population at a high school and requires students to solve problems based on the enrollment data.
- In Algebra I, Topic 8, students must figure out how many miles can be driven in a dune buggy while on vacation with a budget of $75.
- In Algebra I, Topic 16, Student Activity Sheets 2 and 3 use a house floor plan as the context for the problems..
- In Geometry, Topic 15, students use trigonometric ratios to find the height of the flagpole in the courtyard.
- In Algebra II, Topic 15 begins with context that involves graduation money being put towards the purchase of a new car.
The following problems represent the application of key takeaways from Grades 6-8:
- In Algebra I, Topic 3 expands upon 8.F.1 as students define, evaluate, and compare functions. Students look at various situations, create functions, and move into recursively defined functions.
- Students work with proportions and ratios as a key takeaway from grades 6-8 when working with similar figures and dilations in Geometry, Topic 12. Students determine the scale factor (ratio) and a missing coordinate and examine if two figures are similar using proportions.
- Students extend their knowledge of function concepts as students work with linear, exponential, and quadratic functions in Algebra I. In Algebra II, students continue this work with polynomial, rational, logarithmic, and trigonometric functions.
Examples of the materials varying the types of real numbers used across the courses of the series include:
- In Algebra I, Topic 6, Exploring “Rate of Change”, students perform calculations with decimals as they analyze data from a simulation of Hooke’s Law to create a linear model for the data.
- In Algebra I, Topic 13, Student Activity Sheets, students operate on fractions as they apply laws of exponents to simplify expressions involving rational exponents.
- In Geometry, Topic 14, Constructed responses 1 and 2, students perform calculations with decimals and fractions to model and solve real-world problems.
- In Geometry, Topic 22, Guided practice, Page 9, students use irrational numbers to solve an area problem.
- In Algebra II, Topic 1, two problems in the More and Guided Practice sections use a fractional difference in a geometric series.
- In Algebra II, Topic 8, More practice, Pages 9 and 10, students solve problems about joint variation involving mixed numbers and decimals.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series. Overall, connections between and across multiple standards are made in meaningful ways. Each topic provides a Pre-requisite Skills list and an overview of the topic in Topic at a Glance. The Topic at a Glance provides generic connections within each course and throughout the series.
Examples of connections made within courses include the following:
- Algebra I Topic 14 Exponential Functions and Equations connects a number of standards as students create and solve equations in one or two variables (A-CED.1,2) as well as recognize the difference between linear and exponential growth (F-LE) and fit an exponential model to a data set and use models to solve problems (S-ID.6a).
- In Algebra II Topic 5 Polynomial Functions students find zeros using suitable factorizations, if possible, and graph them, connecting A-APR.3, F-IF.7c, and F-IF.8a.
- In Geometry Topic 12 Dilations and Similarity begins by connecting the idea of transformations (G-CO.2) to deciding if two triangles are similar (G-SRT.2). This Topic uses the properties of similarity to prove AA congruence (G-SRT.2) as well as congruence and similarity criteria to solve problems (G-SRT.5). At the end of this topic, students also prove that all circles are similar (G-C.1).
Examples of connections made between the courses include the following:
- Transformations can be found throughout the series. The materials first introduce the idea in Algebra I with translating graphs of functions using the graphing calculator in Topics 15 and 17. Students have extensive work with transformations in Geometry using all transformations (Reflect, Rotate, Translate, and Dilate) on shapes. In Algebra II Topic 3 Transforming Functions, the last lesson in this Topic is titled “Making the algebra-geometry connection,” which makes the algebra-geometry connection between transformations. Transformations are seen in a number of Topics after students extensively work on it in Topic 3 of Algebra II.
- The F-LE standards are connected throughout the series. In Algebra I students compare linear growth and exponential growth in a number of ways and in a number of topics. Students use the idea of linear growth in Geometry to find lines that are parallel and perpendicular. This can be found in Topic 6 which has a lesson called Lines and Algebra. In Algebra II students use the idea from previous coursework to work with logarithms using prerequisite knowledge of exponential functions.
- A-SSE.2 and A-SSE.3.a begin in Algebra I Topics 16 and 18 as students work with operations on polynomials and solving quadratic equations, and they are further developed in Algebra II Topic 6 as students work with polynomial equations.
- In Algebra II Topic 22 Modeling Data focuses on determining an appropriate model for data, interpreting the strength of the relationship between two variables, and making predictions in the context of the problem situation. In the Advice for Instruction, a connection is made between characteristics of function families (F-IF.4) and determining the appropriate model for data in Topic 22. In the Choosing a Model subtopic of Topic 22 students are asked to look at the way data points are spread for the US Census data. Based on prerequisite skills students then determine which shape and/or model is most appropriate. When students reach the Fitting Quadratic Data subtopic in Topic 22 they connect their knowledge of Polynomial Functions Topic 15 from Algebra I and Topic 5 from Algebra II to determine that a parabola is the best fit for the data.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations that the series explicitly identifies and builds on knowledge from Grades 6-8. Materials include and build on content from grades 6-8, however, the content is not clearly identified or connected to specific middle school standards. Although the provided content from Grades 6-8 supports progressions of the high school standards, the Grade 6-8 standards are not identified in either the teacher or student materials.
The following are examples of where the materials build on, but do not explicitly identify standards from Grades 6-8.
- Algebra I Topic 3 Functions includes Prerequisite Skills listed in the Teacher Materials in Prepare Instruction. The list of skills includes “order of operations, operations with rational numbers, domain and range of a function, solving two-step equations by inspection, and plotting points on the coordinate plane.” However, there is no mention in the teacher and/or student materials of where these prerequisite skills can be found in prior content. The teacher is directed in the Deliver Instruction to review independent and dependent variables. These ideas are used to identify domain and range of functions.
- In Algebra I Topic 1 students use prerequisite skills such as perimeter and area of polygons and volume of rectangular solids (7.G.6) in Operations with Polynomials. This topic builds on knowledge from Grades 6-8.
- In Geometry Topic 12 students begin work on an introduction to similarity through dilations. The materials state “The topic dilations and similarity builds on what students have learned about similarity and transformations in middle school in order to generate a precise definition, make connections to transformations, and analyze ways to prove that two triangles are similar.” In the opening question students are expected to determine the distance to place a toy from a flashlight and the height at which to hang the toy such that it casts a particular size shadow on a wall. Students must use proportions (7.RP.2) to solve this problem in relation to similar triangles (G-SRT.5).
- In Geometry Topic 25 students develop formulas for volume and surface area of pyramids and cones. In the Topics at a Glance section of the Advice for Instruction, the materials indicate that “Students should have seen formulas for computing surface area and volume of three-dimensional figures in middle school mathematics.” In this topic students work with the materials to determine the volume of chocolate needed to make a chocolate pyramid and the amount of materials needed to package the chocolate pyramid (Design problem, G-MG.3 using Surface Area, 7.G.6).
- Themes beginning in middle school algebra continue and deepen during high school. As early as grades 6 and 7, students begin to use the properties of operations to generate equivalent expressions (6.EE.3, 7.EE.1). In grade 7, they begin to recognize that rewriting expressions in different forms could be useful in problem solving (7.EE.2). In Algebra I Topic 2 Student Activity Sheet 2 Question 5 students look at a situation where they will build borders of a garden. They look at various ways to set up the two gardens. Throughout the topic students are presented with various situations that ask them to write equivalent expressions both in words and in a numerical representation.
- Students in Grade 8 solve linear equations (8.EE.7) and systems of linear equations (8.EE.8). This concept is built upon in Algebra I Topic 10 as students use this concept to solve real-world problems. In Student Activity Sheet 2 students are given various scenarios to apply their prior knowledge. For example, question 1 involves the context of repairing a gas-powered mower versus buying a new energy-efficient, electric-powered mower. Throughout Student Activity Sheets 3 and 4 students are presented with both systems of equations and inequalities.
- In Algebra II Topic 20 Design and Data Collection in Statistical Studies begins with an overview introducing the idea of sampling. Once in the topic more information is given concerning surveys and sampling. Within the Subtopic Surveys and Sampling, page 12 uses several random samples to produce a dot plot to make decisions about a population. This topic is aligned with 7.SP.2, “Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.”
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The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.
Of the 43 plus standards and 5 plus substandards included in the CCSSM, the materials address 24 of them: N-CN.8,9; N-VM.6-11; A-APR.7; A-REI.8,9; F-IF.7d; F-BF.4c; F-BF.4; F-BF.5; G-SRT.9-11; G-C.4; G-GMD.2; S-CP.8,9; and S-MD.6,7. The materials attend to the full intent of these standards. In general the materials treat these 24 standards as additional content that extends or enriches topics within the unit and do not interrupt the flow of the course. No plus standards were located within the first course of the series, Algebra I.
The following are examples of components of the materials that address the full intent of the plus standards:
- In Algebra II Topic 6 Exploring Other Polynomial Equations, the materials address the fundamental theorem of algebra alongside finding roots of higher order polynomial equations. Students must find one root and then find additional roots using the quadratic formula to identify complex roots (N-CN.9). In the “Check” section, x^2 +1=0 is shown as (x+i)(x-i) (N-CN.8).
- In Algebra II Topic 17 Exploring Using the Inverse Matrix, the materials provide examples of identity matrices for a 2x2 and a 3x3 matrix and ask students to identify what a 4x4 identity matrix would look like based on the provided examples. Students then are asked to describe what they notice about the relationship between the number of rows and columns of identity matrices (N-VM.10).
- In Algebra II Topic 9 Exploring Graphing Rational Functions students use an applet to determine how different parts of rational functions change the graph of a rational function (transformations). Students are also provided opportunities to graph rational functions on the student activity sheets for the topic (F-IF.7d)
- In Geometry Topic 16 Exploring Law of Sines and Law of Cosines students complete proofs of the laws of sines and cosines and use the laws of sines and cosines to solve problems. (G-SRT.10)
- In Geometry Topic 26 Exploring Chocolate Hemispheres students are shown and discuss how the volume of a sphere is derived from a cone and cylinder using Cavalieri’s principle. (G-GMD.2)
- In Algebra II Topic 18 Exploring Permutations and Combinations, students use combinations to determine the number of possible jury members for a trial. (S-CP.9)
Overview of Gateway 2
Rigor & Mathematical Practices
Gateway 2
Criterion 2.1: Rigor
The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. Overall, all three elements of rigor are thoroughly attended to and interwoven in a way that focuses on the needs of a specific standard as well as balancing procedural skill and fluency, application and conceptual understanding.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. There are instances in the materials where students are prompted to use multiple representations to further develop conceptual understanding. In addition, throughout the materials real-world context is used in order to give “concreteness” to abstract concepts, especially when introducing a new topic.
A few examples of the development of conceptual understanding related to specific standards are shown below:
- A-REI.A: In Algebra I Topic 8 students use algebra tiles to solve linear equations. Students check their work using tables and graphs. Algebra II Topic 13 uses graphing technology to introduce logarithmic equations as students examine graphs and tables of logarithmic functions in a real-world context before solving them analytically in the lesson Analytic Techniques.
- A-APR.B: Algebra I Topic 18 begins by using a garden to connect x-intercepts of a graph to zeros of a function. There are a series of questions that students work through in order to connect what is happening graphically with the factored form of a function. This is found again in Algebra II Topic 5 Higher Degree Polynomials. There are graphs and functions (standard and factored form) to show the relationship between zeros and factors of polynomials. Students construct polynomials given the zeros.
- N-RN.1: In Algebra I Topic 13 Laws of Exponents students make tables to see patterns in whole numbers raised to integer exponents, including zero and negative exponents. Students extend this idea to rational exponents by using positive integer exponents and radicals to understand rational exponents.
- F-LE.1: Algebra I Topic 14 begins by introducing students to linear and exponential growth. Students are introduced to different scenarios using fruit flies and fire ants. Materials use the contexts, along with graphs, tables, and functions, to develop students’ conceptual understanding around linear and exponential growth.
- G-SRT.2: Geometry Topic 12 contains applets throughout the lesson that allow students to manipulate triangles in order to further understand triangle similarity through student exploration and guided questioning in Advice for Instruction.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for providing intentional opportunities for students to develop procedural skills. Within the lessons, students are provided with opportunities to develop procedural skills for solving problems. Guided Practice and More Practice sections are included within each lesson. These practice sections are often problems with no context and provide students the opportunity to practice procedural skills when called for by the standards.
Some highlights of development of procedural skills include the following:
- A-APR.1: In Algebra I Topic 16 students multiply binomials to determine the area of rectangles as well as simplify expressions. Students also simplify expressions using polynomial multiplication, addition, and subtraction in the More and Guided Practice sections. In Algebra II Topic 4 students multiply binomials to determine the volume of a rectangular prism as well as simplify expressions using addition, subtraction, multiplication, and division within the More Practice section.
- G-GPE.7: In Geometry Topic 14 students use the distance formula to compute the perimeter of a triangle as well as to determine if the diagonals of a rectangle bisect each other during More Practice. In Topic 21 students use the distance formula to compute the area of a rhombus as well as find the area and perimeter of a hexagon.
- F-BF.3: Students are given examples and applicable activities throughout both Algebra I (Topics 3, 5, 6, 12, 15, and 17) and Algebra II (Topics 3, 4, 9, 10, 12, 13, and 20). For example, Algebra II Topic 3 Making the Algebra-Geometry Connection presents several examples addressing this standard. Student Activity Sheet 4, as well as Practice and Assessment, provide opportunities for students to develop necessary skills.
- G-GPE.4: Geometry Topic 17 Polygons and Special Quadrilaterals explores this standard. Students are given ten examples to view. Example 1 provides definitions and an overview, and Examples 2-10 provide proofs for simple geometric theorems algebraically. Students are then given Student Activity Sheet 4 Practice Problems and Assessment to practice these skills.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Students work with mathematical concepts within a real-world context. There are Topics for which the content of the Topic is framed by a real-world context through the Overview. The context used in the Overview is expanded upon throughout the lessons in the Topic.
Examples of students utilizing mathematical concepts and skills in engaging applications include:
- F-IF.B: The Algebra I Topic 4 Overview introduces students to a skateboarder and his motion. Students use the context of the skateboarder to match his motion to graphs and answer a variety of questions regarding the motion of the skateboarder throughout the lesson. The Topic also uses the idea of elevators and their movement to create graphs and understand various features of the graph. This Topic focuses on interpreting rates of change, and the entire Topic uses a variety of various contexts. In Topic 6 of Algebra I the Constructed Response Assessment problems are examples of applications where students are utilizing mathematical skills to answer various single- and multi-step problems. Students are asked to do things such as find the domain, find and interpret the y-intercept, find a parallel data set, and find the zero and interpret it in relation to the context of the problem. This standard is also found in Algebra II Topic 10 as students engage with the problems in the Topic using sets of data modeled by square root equations to further develop the mathematical skills for examining and identifying the features of graphs.
- G-SRT.8: Geometry Topic 15 Indirect Measurement begins with the student council of a school finding the height of the flagpole in the school courtyard. The lesson takes students through different strategies to find the height. The last question in the lesson uses an airplane to find the angle of depression. There are a number of application problems in the practice and assessment.
- S.ID.2: Algebra II Topic 19 has a number of application situations for students to use surveys and sampling. The Overview for the Topic uses real-world context by doing a survey to see if Americans believe life exists beyond earth. This application is used throughout in order to make sense of key terms. The questions in the Advice for Instruction provides the teacher with strategies to support students to work through the application in more meaningful ways. The entire Topic has a variety of application problems in Exploring, Practice, More Practice, Assessment, and Student Activity Sheets.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for the three aspects of rigor being balanced with respect to the standards being addressed. The structure of the materials lends itself to balancing the three aspects of rigor.
Each Topic includes an Overview, Exploring, Practice, Assessment, and Student Activity Sheets.
- The Topic Overviews provide a focal point for students to begin thinking about the Topic. They allow for students to relate the topic to a real-world application and/or prior knowledge. This gives students an opportunity to develop conceptual understanding through applications and/or prior knowledge. For example, in Algebra II Topic 4 the teacher is provided with the pieces of a puzzle. In the opening to the lesson teachers state “suppose you buy 100 pencils for $25.” The teacher is then presented with several framing questions “What is the cost of each pencil? How do you know? What operation did you perform to find the answer? If you bought x pencils for $25, what expression would represent the cost of each pencil?”
- The Exploring section focuses on developing conceptual understanding, in context and/or by using applets. Students are given the tools to build their procedural skills throughout as algorithmic steps are connected to the concepts in this section.
- Practice has Guided Practice and More Practice for students. There are a variety of types of problems (multiple choice, multiple select, true or false, etc) with a focus on conceptual understanding and procedural skills. Students can get hints and immediate feedback if their answer is correct. If it is incorrect, students receive a statement/question to help direct their thinking.
- Assessment has two parts, Automatically Scored and Constructed Response. Automatically Scored includes Multiple Choice and Short Answer. This section has questions that require conceptual knowledge, procedural skills, and application of the Topic.
- Student Activity Sheets follow the online instruction but include additional procedural skill and application problems.
In addition to this, there are MARS tasks throughout that focus on conceptual understanding and application.
The following are examples of balancing the three aspects of rigor in the instructional materials:
- Algebra I Topic 5 Moving Beyond Sloping Intercept (S.ID.7) has students study data from a table of a skateboarder and their distance traveled during a set of skateboard drills. They use the data table to match the graph of the motion detector data to the path created by moving the computerized skater on the app provided. Students discuss the two parameters necessary to match the graph and develop understanding around steepness (slope) of the line and the constant (intercept). Students use this knowledge to do more procedural skills around standard form and point-slope form. Throughout the Topic, students are given real-world context to explore all concepts in this Topic.
- Geometry Topic 17 Polygons and Special Quadrilaterals (G-GPE.4) includes examples that review key concepts from previous topics and subtopics. In the topic, the meaning of coordinate proof is given and then stepped through the idea of special quadrilaterals. Students have a series of questions posed to answer (developing conceptual understanding), examples of the process to do a coordinate proof (developing procedural skills), and more questions to check their understanding throughout the lesson on Coordinate Proofs. Students then work in small groups to write a coordinate proof showing that diagonals of a rhombus are perpendicular. After completing these things, students can do the Guided Practice and More Practice to master the skills just learned.
- Algebra II Topic 2 Understanding Inverse Relations (F-BF.4.a) begins with a real-world context to introduce the idea of inverse relations. Throughout the topic, students are given contextual situations to bridge the idea of functions and their inverses together. The use of tables, graphs, and equations are all used throughout in order for students to understand the idea of inverse. Students are also given the opportunity to compare a function and its inverse as well as practice the skill of finding an inverse from a function. Students are also given real-world context to answer questions which require a conceptual understanding of inverse relations.
Criterion 2.2: Math Practices
The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials reviewed meet the expectations for the development of overarching, mathematical practices; reasoning and explaining; modeling and using tools; and seeing structure and generalizing.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). The instructional materials develop both MP1 and MP6 to the full extent of the MPs. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. Teachers are given guidance on materials in the Advice for Instruction provided in each Topic. Topics that include a MARS Task list MPs used within the task in the Advice for Instruction. The Advice for Instruction also provides teachers with guidance to foster discussion throughout the materials. This discussion often stresses accurate vocabulary used to increase precision of mathematical language. Emphasis is placed on using units of measure and labeling axes throughout the series. Making sense of answers within the context of a problem is also emphasized. Students persevere in problem solving in lessons through the many real-world application scenarios.
- In Geometry Topic 14 Block 5 MARS task students make sense of problems and persevere in solving them (MP1) to determine angles and lengths of pieces of wood of a wooden garden chair using similarity and Pythagorean Theorem (G-SRT.8).
- In Algebra I Topic 1 Student Activity Sheet 1 the first question has students look at three linear graphs and asks them “How are the three graphs similar? How are the three graphs different?” The students analyze the graphs, make conjectures, and plan their strategy. As the teacher directs the lesson, the teacher is told to encourage students’ thinking and emphasize the importance of precise communication by helping students use precise language (MP6) such as slope, how steep, increasing, decreasing, and line versus segment (F-IF.6).
- Algebra I Topic 8 Student Activity Sheet 4 Question 31 introduces a babysitting situation through Maggie who charges $15 an hour for babysitting. The students look for the possible numbers of hours she can babysit to make enough money for the bag she wants to buy without having to pay taxes on her earnings. In this problem, students are asked to first understand the meaning of the problem, analyze the information, make a conjecture, and try to find an answer. Students need to check their answers and see if their answer makes sense (MP1) (A-CED.3).
- In Algebra II Topic 18 students are presented with an overarching scenario. In this scenario, Mr. Jones witnesses a crime and gives a witness description. In the scenario, students reason through multiple probability concepts for each portion of Mr. Jones’ account. “How likely is it that Rob is the bad guy? Mr. Jones indicated that the license plate had 6 non-repeating letters.” In this case, the students (playing the role of Rob’s attorney) will have to use the fundamental counting principle to determine how likely it is to have a license plate with non-repeating letters. Students then use basic probability concepts to determine probabilities of events occurring together and/or occurring given that another event occurred first. Lastly, students use the normal distribution to find the percentage of people in Arresta that have a height of 6 feet or more. Students persevere through multiple nuances of the problem to finally determine their solution to whether the jury comes back with a guilty or not guilty verdict (MP1) (S-ID.4).
- In Algebra II Topic 1 Exploring Arithmetic Sequences and Series students are first presented with an auditorium seating scenario. In this scenario, students are told that the first row has 20 seats and each subsequent row has 3 seats more than the row in front of it. Students are given a table to complete in Student Activity Sheet 2 question 3. From here, the students are asked to determine the constant difference of the linear function that models this situation and also determine the domain and range. The student then writes a precise arithmetic sequence that will help them predict the 21st term in the sequence (MP6) (A-SEE.4).
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to reason abstractly and quantitatively, and students are expected to construct viable arguments by justifying along with a few opportunities to critique the reasoning of others in each course. Support and guidance is provided for teachers in the Advice for Instruction for each lesson to assist teacher development of these MPs although the MPs are not explicitly listed in the Advice for Instruction with the exception of the MARS Tasks. MARS Tasks identify specific MPs used within each task in the Advice for Instruction.
- In Algebra I Topic 3 Functions, Exploring Modeling with Functions begins with the real-world application of a soccer team selling roses as a fundraiser. Students are asked at the beginning of the lesson “What a successful project looks like” and “How will the factors in the list influence the success of the project?” Students think about successful money-making ventures and discuss the factors that make the ventures successful. Students are then given amounts and pricings for two shops to determine which shop would be the best deal for them. Students are then asked to determine costs for ordering a specific numbers of roses from each shop and explaining their reasoning to the class. In this scenario students formulate their own reasoning and explain the evidence and/or prompts that led to their reasoning, illustrating both MP2 and MP3 (F-IF.1).
- In the Geometry Topic 15 MARS Task students reason abstractly (MP2) about whether a particular triangle is a right triangle by finding the side lengths of neighboring triangles and then using the Pythagorean Theorem to draw a conclusion. Students must also explain how they decided the triangle was a right triangle (MP3) (G-SRT.8, G-SRT.5).
- One example where students are given the opportunity to critique the reasoning of others (MP3) can be found in Algebra II Topic 21 Student Activity Sheet 5 Problem 8. Students are shown a graph of a cotangent function along with a student’s description of the function. Students are directed to critique the student’s description of the graph of the cotangent function.
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The instructional materials reviewed for Agile Mind Traditional series meet the expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5).
The materials fully develop MPs 4 and 5 as students build upon prior knowledge to solve problems, create and use models within many lessons, and choose and use appropriate tools strategically across the courses. The materials pose problems connected to previous concepts and a variety of real-world contexts. Students are provided meaningful real-world problems in which to model with mathematics and use tools.
Examples where students model with mathematics include:
- In Algebra I, Topic 6, Exploring, students examine a graph of shoe size versus height of a person. Students create a function to model the information provided in the graph. Students were able to use previous knowledge to show a correlation between each of the figures and thus draw a trend line. The students formulated the problem (with some help from the book), then were asked to compute the trend line that would correlate that data and lastly students checked their work and reported out. This process is present for multiple scenarios within the guided practice and more practice sections.
- In Algebra I, Topic 15, MARS Task: Functions, students “model each of two subsets of a set of points on a scatterplot. Students must go beyond simple visual inspection of a graph to sort the set into two subsets and justify their sorting by applying their knowledge of fundamental characteristics of different function families.” In this activity, students must write a linear function to represent the scatterplot and determine a non-linear model for the rest of the points in the scatterplot. Students must verify their solutions with their partner and report their findings.
- In Algebra II, Topic 21, Student Activity Sheet 4, students are given data representing the number of hours of daylight in Tallahassee, Florida for the year 1998. Students are asked to “Make a scatterplot of these data using your graphing calculator. What type of function do you think would model these data? Do you think these data are periodic?” Later, students are asked, “What trigonometric function would you use to model the data?” and then “What is the period of the graph?” “What is the amplitude of the sinusoidal graph?” “Is the graph shifted horizontally and/or vertically from the parent function y = sin x? If so, by how much is it shifted?” “Transform the parent function, y = sin x, to fit the data.” Finally, students are asked to “Use your model to find the days when Tallahassee had more than 12 hours of daylight.” Through this set of problems, students apply prior knowledge to new problems, identify important relationships and map relationships with tables, diagrams, graphs, rules, draw conclusions as they pertain to a situation, create, and use models.
Examples where students choose and use appropriate tools strategically include:
- In Geometry, Topic 8, Block 4, Advice for Instruction states, “Encourage students to use tools to help them make sense of the problem of dividing a triangle into sixths. Make Patty Paper, rulers, protractors, dynamic geometry software (optional) and scissors available to students. Give students enough time to really try to answer this question.”
- In Geometry, Topic 11, Block 1, Advice for Instruction, students follow a paper-and-pencil activity with a construction activity in which they are to “use tools of their choice.” Later in the same block, in Technology tip, pages 6-7, Exploring “Congruent segment and angle bisector constructions,” students choose between compass and straightedge or an online construction tool.
- In Algebra II, Topic 20, Student Activity Sheet 2, Question 10, students design and carry out a simulation. They choose from a variety of tools to carry out the simulation, including a coin, a random number table, a random number generator, or a statistical software package.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, support is present for the intentional development of seeing structure and generalizing.
- In Geometry Topic 13 Geometric Mean students define the geometric mean. Students receive instruction on the connection between the geometric mean and right triangles. They work a puzzle to apply what they have learned and determine the difference between the arithmetic mean and the geometric mean. They are asked to make a general statement about when the arithmetic mean and the geometric mean of two numbers are the same. This provides students with the opportunity to see structure and generalize to a larger idea (MP7).
- In Algebra II Topic 19 Conditional Probability and Independence students are asked to determine the conditional probability that a British ship is armed given that it appears armed. This question arises from British ships’ need to deter pirates. They then use their conclusions from Exploring Conditional Probability to help determine calculations from the French ship data to show that the event of selecting an armed French ship is not dependent on the event of selecting a French ship that appears to be armed. This allows the students to use requisite knowledge to make generalizations from the sample data (French Ships) to the population (British Ships) (MP7).
- In Geometry Topic 21 students make estimates about the area of the model and then refine that estimate by using different scales for estimating the area through different geometric figures through repeated reasoning (MP8).
- In the Algebra I Topic 4 MARS Task Differences students look for patterns and express regularity in repeated reasoning by completing a table (MP8). By completing the table, students are able to notice patterns in a sequence and determine the 7th and 8th term in the sequence. Students must also use their completed table, along with a table of expressions, to look for patterns in both tables to determine the coefficients of the expressions given in the second table.
Overview of Gateway 3
Usability
Criterion 3.1: Use & Design
The instructional materials reviewed for Agile Mind Traditional series meet expectations that the materials are well designed and take into account effective lesson structure and pacing. Overall, materials are well-designed, and lessons are intentionally sequenced. Students learn new mathematics in the Exploring section of each Topic as they apply the mathematics and work toward mastery. Students produce a variety of types of answers including both verbal and written answers. The Overview for the Topic introduces the mathematical concepts, and the Summary highlights connections within and between the concepts of the Topic. Manipulatives such as algebra tiles and virtual algebra tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.
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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.
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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.
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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.
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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.
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The instructional materials reviewed for Agile Mind Traditional series have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials are clear and consistent between Topics within a grade-level as well as across grade-levels. Each piece of a Topic is clearly labeled, and the explorations include Page numbers for easy reference. Problems and Exercises from the Practice, Assessment, and Activity Sheets are also clearly labeled and consistently numbered for easy reference by the students. There are no distracting or extraneous pictures, captions, or "facts" within the materials.
Criterion 3.2: Teacher Planning
The instructional materials reviewed for Agile Mind Traditional series meet expectations that materials support teacher learning and understanding of the standards. The instructional materials provide Framing Questions and Further Questions that support teachers in delivering quality instruction, and the teacher’s edition is easy to use and consistently organized and annotated. Different sets of interactive, print, and video essays provide teachers with adult-level explanations or examples of advanced mathematics concepts to help them improve their own knowledge of the subject. Although each Topic contains a list of Prerequisite Skills, this list does not connect any of the skills to specific standards from previous grade levels, so the instructional materials partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.
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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?”
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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]”
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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.
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The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum. In the course materials for Algebra I, Geometry, and Algebra II, the specific reference to the Standards is the following statement in the Plan the Course materials: “Alignment to standards, … you will find correlations from Agile Mind topics to your state learning standards. These alignments can be found in Course Materials.” There are no specific references within the online lesson materials as to the standards that are being taught for the courses. A Scope and Sequence is provided where the standards for each lesson are listed for each Topic.
Within Professional Support, Practice Standards Connections is provided. Also, the materials include a table for each Standard for Mathematical Practice that lists examples of where the MPs are used within the course. “The citations below are examples from the Algebra II program that show how the materials provide students with ongoing opportunities to develop and demonstrate proficiency with the Standards for Mathematical Practice.” Teachers are able to make connections between the standards being taught and the activities and instruction for the lesson.
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The instructional materials reviewed for Agile Mind Traditional series provide a list of lessons in the teacher's edition, cross‐referencing the standards covered and providing an estimated instructional time for Topics and Blocks. For each course, the materials provide a Scope and Sequence document which includes the number of Blocks of instruction for the duration of the year, time in minutes that each Block should take, and the number of Blocks needed to complete each Topic. The Scope and Sequence document lists the CCSSM addressed in each Topic, but there is no part of the materials that aligns Blocks to specific content standards. The materials also provide Alignment to Standards in the Course Materials which allows users to see the alignment of Topics to the CCSSM or the alignment of the CCCSM to the Topics. The Deliver Instruction section contains the Blocks for each Topic. The Practice Standards Connections, found in Professional Support, gives examples of places in the materials where each MP is identified.
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The instructional materials reviewed for Agile Mind Traditional series do not contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
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The instructional materials reviewed for Agile Mind Traditional series do not contain explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials. There is a Professional Essays section which addresses a broad overview of mathematics and clips of teachers using Agile Mind in Algebra I, Geometry, and Algebra II as discussed in indicator 3h.
Criterion 3.3: Assessment
The instructional materials for Agile Mind Traditional series partially meet exceptions that materials offer teachers resources and tools to collect ongoing data about students progress on the Standards. Opportunities for ongoing review and practice, and feedback occur in various forms. Standards are identified that align to the Topic; however, there is no mapping of Standards to items. There are opportunities for students to monitor their own progress, and there are assessments that explicitly identify prior knowledge within and across grade levels. The materials include opportunities to identify common misconceptions, and strategies to address common errors and misconceptions are found in Deliver Instruction topics.
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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.”
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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.”
Indicator {{'3o' | indicatorName}}
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.
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The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for assessments clearly denoting which standards are being emphasized.
The pre-made assessments provided in the Assessment section align to the standards addressed by the Topic, but the individual items are not clearly aligned to particular standards. The set of standards being addressed by a Topic can be found in the Scope and Sequence document or in Course Materials through Alignment to Standards. The MARS Tasks also do not clearly denote which CCSSM are being emphasized.
Agile Assessment is an optional resource that can be licensed along with the Agile Mind Traditional Math series, and Agile Assessment allows educators to create their own assessments by selecting from a repository of items aligned to standards and level of difficulty. Reports from assessments created with Agile Assessment denote which standard is being assessed.
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The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The MARS Tasks and selected Constructed Response items in Algebra II are accompanied by rubrics aligned to the task or item that show the total points possible for the task and exactly what students need to do in order to earn each of those points. The remainder of the Constructed Response items in Algebra II, along with all of the Constructed Response items in Algebra I and Geometry, are accompanied by complete solutions, but rubrics aligned to these Constructed Response items are not included. For both the MARS Tasks and the Constructed Response items, alternate solutions are provided when appropriate, but sufficient guidance to teachers for interpreting student performance and suggestions for follow-up are not provided with most of the MARS Tasks or the Constructed Response items. In Algebra I, there are four Constructed Response items that are accompanied by a professional essay titled “Learning from Student Work”, and Algebra I and Geometry each include a MARS Task that is accompanied by a professional essay that provides guidance to teachers for interpreting student performance and suggestions for follow-up.
Indicator {{'3q' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series offer opportunities for students to monitor their own progress. Throughout the Exploring, Practice, and Automatically Scored Assessment sections, students get feedback once they submit an answer, and in that moment, they can adjust their thinking or strategy. Goals and Objectives for each Topic are not provided directly to students, but they are given to teachers in Prepare Instruction.
Students can also monitor their progress on assignments and quizzes assigned by their teacher from the Agile Mind Traditional courses. There is a set of reports for students that appear on their dashboard about active assignments and quizzes from that day, there is another set of reports in the student’s Report area from which students can view data on all the assignments they have completed throughout the year. These reports allow students to monitor their progress and learning related to the topics in the course.
Criterion 3.4: Differentiation
The instructional materials for Agile Mind Traditional series partially meet expectations that materials support teachers in differentiating instruction for diverse learners within and across grades. Activities provide students with multiple entry points and a variety of solution strategies and representations. The materials also provide strategies for ELL and other special populations, but they do not provide strategies for advanced students to deepen their understanding of the mathematics. Grouping strategies are designed to ensure roles for each group member.
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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.
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The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for providing teachers with strategies for meeting the needs of a range of learners. Overall, the instructional materials embed multiple visual representations of mathematical concepts where appropriate, include audio recordings in many explorations, and give students opportunities to engage physically with the mathematical concepts.
However, the instructional notes provided to teachers do not consistently highlight strategies that can be used to meet the needs of a range of learners. When instructional notes are provided to teachers, they are general in nature and are intended for all students in the class, and they do not explicitly address the possible range of needs for learners. For example, in Algebra II, Topic 6, Block 3, Deliver Instruction states, “To save time, break the classroom into three sections. Have one section solve the first equation, another the second, and the last section the third equation. Give each section time to solve their equation and check their work with each other, as well as time to interpret their graph and number of solutions and to pick a person to present. Have a member from each section come up and present the work.”
In some explorations, teachers are provided with questions that can be used to extend the tasks students are completing, which are beneficial to excelling students. The Summary for each Topic does not provide any strategies or resources for either excelling or struggling students to help with their understanding of the mathematical concepts in the Topic. For struggling students, teachers are occasionally provided with strategies or questions they can use to help move a student’s learning forward. For example, in Geometry, Topic 26, Block 1, Advice for Instruction states, “To differentiate instruction, you can give constraints to the spheres that students create. For example, you can limit the side lengths to whole numbers. Students may have individual modifications that allow them to use a calculator."
Indicator {{'3t' | indicatorName}}
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.
Indicator {{'3u' | indicatorName}}
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.”
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The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. The problems provided in the materials are on course level, and the materials are designed to assign most of the problems to all students. However, there are a few problems that are on course level and not assigned to all students, and these problems could be used for advanced students to investigate mathematics content at greater depth. Examples include:
- In Algebra I, Topic 4, the MARS Task “Differences” does not have to be assigned to all students at the completion of the Topic, and could be assigned to advanced students.
- In Geometry, Topic 18, optional Block 5, students investigate another curve of constant width as they study Reuleaux triangles.
- In Geometry, Topic 23, optional Block 5, students investigate orthographic and isometric drawings.
Indicator {{'3w' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series provide 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.
Indicator {{'3x' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series provide opportunities for teachers to use a variety of grouping strategies. The Deliver Instruction Lesson Activities include suggestions for when students could work individually, in pairs, or in small groups. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions to ensure the involvement of each group member. For example, in Algebra II, Topic 10 Block 2 Deliver Instruction teachers are told to “Have student pairs solve the equation algebraically, then use the solution to determine how fast Chloe runs and how long her workout takes. [SAS 2, question 17] Ask for student volunteers to share their processes. Use page 10 as needed to verify responses.”
Indicator {{'3y' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series do not encourage teachers to draw upon home language and culture to facilitate learning. Questions and contexts are provided for teachers in the materials, and there are no opportunities for teachers to adjust the questions or contexts in order to integrate the home language and culture of students into the materials to facilitate learning.
Criterion 3.5: Technology Use
The instructional materials for Agile Mind Traditional series are web-based and platform neutral but do not include the ability to view the teacher and student editions simultaneously. The materials embed technology enhanced, interactive virtual tools, and dynamic software that engage students with the mathematics. Opportunities to assess students through technology are embedded. The technology provides opportunities to personalize instruction; however, these are limited to the assignment of problems and exercises. The materials cannot be customized for local use. The technology is not used to foster communications between students, with the teacher, or for teachers to collaborate with one another.
Indicator {{'3aa' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series are web-based and compatible with multiple internet browsers (Chrome, Firefox, and Internet Explorer). In addition, the materials are “platform neutral” and allow the use of tablets with ChromeOS, Android, or iOS operating systems, and students can complete assignments on smartphones.
However, the navigation between the online student and teacher materials and resources are cumbersome and time consuming. The online interface makes it difficult to compare the student and teacher materials since they cannot be seen in their entirety simultaneously. Teachers can review the printed, spiral-bound teacher materials while viewing the online curriculum projected in class (and what the student also sees when they log into the system).
Indicator {{'3ab' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. All Practice and Automatically Scored Assessment questions are designed to be completed using technology. These items cannot be edited or customized.
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The instructional materials reviewed for Agile Mind Traditional series include few opportunities for teachers to personalize learning for all students. Within the Practice and Assessment sections, the teacher can choose which problems and exercises to assign students, but these problems and exercises cannot be modified for content or wording from the way in which they are given. Other than being able to switch between English and Spanish in My Glossary, there are no other adaptive or technological innovations that allow teachers to personalize learning for all students.
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The instructional materials reviewed for Agile Mind Traditional series cannot be easily customized for local use. Within My Courses, there are no options for modifying the sequence or structure of the Topics or any of the sections within the Topics.
Agile Assessment is an optional resource that can be licensed along with the Agile Mind Traditional series, and Agile Assessment allows educators to create their own assessments by selecting from a repository of items aligned to standards and level of difficulty.
Indicator {{'3ad' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series provide few opportunities for teachers and/or students to collaborate with each other. Under My Agile Mind, teachers can communicate with students through the Calendar and Score and Review. There are no opportunities for teachers to be able to collaborate with other teachers.
Indicator {{'3z' | indicatorName}}
The instructional materials reviewed for Agile Mind Traditional series integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Given the digital platform of the materials, the inclusion of interactive tools and virtual manipulatives/objects helps to engage students in the MPs in all of the Topics, and the use of animations in all of the Topics provides for some examples as to how the interactive tools and virtual manipulatives can be used.
Report Overview
Summary of Alignment & Usability for Agile Mind | Math
Product Notes
Upon completing the review of the High School Integrated series from Agile Mind, EdReports.org determined that revisions and enhancements made within the Integrated series had also been made to the materials for the Traditional series and Grades 6-8 from Agile Mind. As a result, EdReports.org revised the reports for the Traditional series and Grades 6-8 to reflect the revisions and enhancements that have been made to the materials.