High School - Gateway 1

The instructional materials reviewed for the Walch Traditional Florida series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials include few instances where all aspects of the standards are not addressed across the courses of the series.

The following are examples for which the materials attend to the full intent of the standard:

• N-CN.3.7: In Algebra 2, Lesson 1A.6, students recall the imaginary unit i and how it applies to the Fundamental Theorem of Algebra in Key Concepts and Instruction. In Scaffolded Practice and Guided Practice, students solve quadratic equations with complex solutions.
• A-REI.2.3: In Algebra 1, Lesson 1.10, students solve linear equations. Throughout the lesson, students solve equations with variables on the same side of the equal sign, variables on opposite sides of the equal sign, and use the distributive property. Students also solve equations with coefficients represented by letters as seen in Practice Problems 9 and 10.
• F-IF.3.9: In Algebra 1, Lesson 2A.12, students compare linear functions represented by tables, equations, and graphs. Students compare y-intercepts and rates of change of linear functions. In Lesson 2A.13, students compare rates of change and y-intercepts of exponential functions in multiple ways. In the Problem-Based Task of Lesson 2A.13, students compare two medical procedures used to measure how well a person’s kidneys are functioning. Students compare the initial value of each procedure, as well as the rate of decay to determine which has a faster rate of decay.
• F-LE.2.5: In Algebra 2, Lesson 3A.4, students investigate the exponential growth of a savings account. Students interpret the meaning of each term of the function in context to the problem. Throughout the lesson, students work with population growth, half-life of radioactive material, and cooling of liquids. Students identify the initial value and relate its meaning to its context. Students create and compare equations to model a savings account in the Problem-Based Task in Lesson 3A.5. Students identify domain constraints and the outstanding balance after t months.
• G-SRT.3.7: In Geometry, Lesson 2.11, students explore the relationship between the sine and cosine of complementary angles. Through the Warm-Up, Scaffolded Practice, and Guided Practice, students explore sine and cosine as complements. Students determine the best location for a garden, based on the amount of sunlight it will receive, in the Problem-Based Task in Lesson 2.11. Students apply sine, cosine, and the Pythagorean Theorem in 30-60-90 degree triangles to defend their findings.
• G-C.1.1: In Geometry, Lesson 5.1, students extend their knowledge of similarity of triangles to prove similarity of circles. In the Problem-Based Task of Lesson 5.1, students prove all circles are similar. Students identify circular objects in the classroom and apply relationships between the parts of the circle (circumference and diameter) to prove similarity.
• S-IC.1.1: In Algebra 2, Lesson 4.4, students explore if a number of possible samples exist even for a small population. In the Guided Practice, students examine variability in sampling and assess a set of data representing the frequency of the most requested songs over a six-year time period in the Problem-Based Task. Students choose a sample of three songs from the six years and compare the mean and standard deviation of the sample to that of the population.

The materials attend to some aspects, but not all, of the following standards:

• G-CO.4.12: In Geometry, Lesson 1B.9 (copying segments and angles), Lesson 1B.10 (bisecting segments and angles), and Lesson 1B.11 (constructing perpendicular and parallel lines), students make formal geometric constructions using a straightedge and compass. In Lesson 1B.9, the Introduction and Key Concepts references utilizing patty paper, but students do not use this tool throughout the lesson. Other tools or methods for creating formal geometric constructions are not referenced in the materials.
• G-MG.1.1: In the Introduction of Geometry, Lesson 3.3, there is an example of cutting a carrot in half, resulting in either a circle or oval, depending upon the type of cut. No other opportunities were found where the materials use geometric shapes, their measures, and their properties to describe objects.

The instructional materials reviewed for the Walch Traditional Florida series, when used as designed, meet expectations for allowing students to spend the majority of their time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs, and careers. Examples of how the materials allow students to spend the majority of their time on the WAPs include:

• In Algebra 1, Unit 2A, Lessons 7-9, students use appropriate vocabulary to describe functions. Students analyze a function in terms of its graph and the context of the situation the graph depicts. In Lesson 2A.8, students calculate the average rate of change and compare rates of change over specific intervals (F-IF.2.6, F-LE.1.1a).
• In Algebra 1, the pacing guide recommends 25 days for Unit 2B, and of those days, 20 address WAPs from A-REI, solving systems of equations and inequalities, and F-BF.1.1 and F-LE.1.1 on creating equations and distinguishing between linear and exponential functions.
• In Geometry, Unit 2, Lessons 5-9, students extend their knowledge of triangle similarity statements to include Side-Angle-Side (SAS) and Side-Side-Side (SSS). Students use these similarity statements to prove the Pythagorean Theorem (Lesson 2.5). Students work with triangles related to this example and expand on their understanding of similar triangles, specifically the relationship between angle bisectors and ratio segments. In Lesson 2.9, students build an understanding of ratios of lengths of sides within right triangles (G-SRT.2.4,5).
• In Geometry, Unit 1B, Lessons 3-6, students prove each of the theorems represented in G-CO.3.10. For example, students prove measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side, and half the length and the medians of a triangle meet at a point.
• In Algebra 2, the pacing guide recommends 27 days for unit 3A, and of those days, 18 address WAPs from F-IF (graphing trigonometric, exponential, and logarithmic graphs). Students also interpret and identify key features of these graphs.
• In Algebra 2, Lesson 3.9, students interpret the slope and intercept of a linear model in the context of data (S-ID.3.7).

The instructional materials reviewed for the Walch Traditional Florida series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from grades 6-8.

Examples where the materials regularly use age-appropriate contexts with various types of real numbers include:

• Algebra 1: In Lesson 2A.10, students organize a fundraising concert for their community. Students identify combinations of numbers, including decimals, to produce equivalent values. Throughout Lesson 2A.10, students work with integers and rational values.
• Algebra 1: In Lesson 5.2, students create a quadratic model to determine the path of a basketball. Students explore the reasonableness of decimal values related to measurement in feet.
• Geometry: In Lesson 3.3, students describe the building of a new observatory planned for a local college in terms of its basic geometric shapes. They also determine the volume of the dome of the building and other parts of the building.
• Geometry: In Lesson 2.13, students apply their knowledge of the Pythagorean Theorem and trigonometric ratios to calculate the distance from the roof a building to a windowsill where firefighters will need to descend by rope to enter the building. Students encounter decimal values to the thousandths place and make sense of the solution in context. Students work with rational and irrational numbers in this context.
• Algebra 2: In Lesson 3B.12, students use graphing technology to create exponential, linear, and quadratic functions for data sets representing the growth in users of a popular social media website. Students work with data sets representing millions of users, and as students choose a model, the values are written in scientific notation.
• Algebra 2: In Unit 2, Problem-Based Task 2.5, students determine the best wax for a surfboard. The method involves placing a block on the board and lifting the end. The sooner the block slides, the better the wax. Students work with decimal values and angles as they discover the angle that produces the best wax as seen by the movement of the block.

Examples where the materials regularly provide opportunities for students to apply key takeaways from grades 6-8 include:

• Algebra 1: In Lesson 2A.7, students apply their understanding of graphs (8.F.2.4) and begin to more formally describe various features. Students interpret key features from graphs and tables, as well as sketch graphs when given a verbal description.
• Geometry: Students apply proportions and ratios as key takeaways from grades 6-8 as they find arc lengths and areas of sectors in Lessons 5.8 and 5.9. Students use the definition of a sector as the portion of a circle bounded by two radii and their intercepted arc to create and solve a proportion to find the area of the sector.
• Algebra 2: In Lessons 1A.1-3, students find sums and differences of complex numbers by applying their understanding of combining like terms (8.EE.3.7b) and the Commutative Property of Addition (6.EE.1.3).

The instructional materials reviewed for the Walch Traditional Florida series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards.

Examples where the materials foster coherence through meaningful mathematical connections in a single course include:

• Algebra 1: In Lesson 1.3, students create linear equations in context (A-CED.1.1). In Lesson 1.6, students extend creating linear equations to creating and graphing linear equations in two variables (A-CED.1.2). In Lesson 1.10, students solve linear equations, and in Lesson 2B.6, students continue the progression of solving linear equations as they solve systems of linear equations through substitution and elimination (A-REI.3.5).
• Algebra 2: In Lessons 1B.3-5, students add, subtract, multiply, and divide rational expressions (A-APR.4.7). Students extend their understanding of rational expressions as they solve rational equations in Lessons 1B.6 (A-REI.1.2). In lessons 1B.8, students graph rational functions (F-IF.3.7).
• Geometry: In Lessons 1A.6 and 1A.7, students encounter properties of congruence within geometric figures (G-CO.2.6). In Lesson 2.4, students work with similar figures (G-SRT.2.4). In Lessons 2.6 and 2.8, students determine how side lengths of similar figures are proportional instead of congruent (G-SRT.2.5).

Examples where the materials foster coherence through meaningful mathematical connections throughout the series include:

• In Algebra 1, Lessons 2B.6 and 2B.7, students solve systems of linear equations (A-REI.3.7). In Algebra 1, Lessons 4.19 and 4.20, students extend that knowledge to solving systems of linear and quadratic equations. In Algebra 2, Lessons 1B.10-12, students review solving systems of equations graphically and extend that knowledge to rational and radical equations.
• In Geometry, Lesson 2.7, students extend their understanding of the Pythagorean Theorem as they prove it through the application of similarity (G-SRT.2.4). In Algebra 2, Lesson 2.5, students apply the Pythagorean Theorem to trigonometric identities.
• Transformations are found throughout the series. In Algebra 1, Lesson 2B.4, students transform linear and exponential functions. In Algebra 1, Lessons 5.17 and 5.18, students transform quadratic functions. In Geometry, Lessons 1A.4 and 1A.5, students use rotations, reflections, and translations in their work with transformations. In Algebra 2, Lesson 3B.7, students work with transformations of parent functions.

The instructional materials reviewed for the Walch Traditional Florida series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The standards from Grades 6-8 are explicitly identified in the Introduction and Key Concept, and the materials build on prerequisite knowledge. Examples where the materials identify standards and build on them include:

• In Algebra 1, lesson 1A.1, students translate verbal expressions into algebraic expressions and use order of operations (6.EE.2) to simplify algebraic expressions (A-SSE.1.1).
• In Algebra 1, Lesson 5.1, students extend the properties of integer exponents (8.EE.1.1) when simplifying expressions with rational exponents (N-RN.1.2).
• In Algebra 1, Lesson 2A.8, students use their previous learning of reading and interpreting data from charts and tables (6.EE.3.9) and understanding slope (8.EE.2.5) to calculate average rate of change of a function over a specified interval given a graph. Students also compare rates of change over various intervals (F-IF.2.6).
• In Geometry, Lesson 2.1, students apply operations with fractions (7.NS.1.1,2) and operations with fractions and decimals (7.EE.2.3) to calculate scale factors for similar figures (G-SRT.1.2).
• In Geometry, Lesson 2.10, students apply the Pythagorean Theorem (8.G.2.7) and ratios (7.RP.1.3) to solve problems involving trigonometric ratios and right triangles (G-SRT.3.6).
• In Geometry, Lesson 5.8, students find the circumference of circles (7.G.2.4) to find arc lengths (G-C.2.5).
• In Algebra 2, Lesson 3.1, students find the median and quartiles of a data set (6.SP.2.5) to construct box plots (S-ID.1.1).