High School - Gateway 1

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.

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.

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.

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.

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.