The instructional materials for the Walch Traditional Florida series partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials, across the series, do not develop conceptual understanding for several standards that address conceptual understanding.

In addition, the Program Overview for each course contains Conceptual Activities, which lists resources by unit, but these resources often do not address conceptual understanding. For example, in Algebra 2, Computations with Complex Numbers is a Conceptual Activity where students develop procedural skills with operations on complex numbers.

Examples where the materials do not develop conceptual understanding for specific standards across the series include:

• N-RN.1.1: In Algebra 1, Lesson 2A.1, students evaluate and simplify expressions with rational exponents. In Algebra 2, Lesson 1B.1, students practice evaluating and simplifying expressions with radicals and rational exponents by applying the properties of integer exponents. Students do not develop understanding by explaining how the definition of the meaning of rational exponents extends from the properties of integer exponents.
• A-APR.2.2,3: In Algebra 1, Lesson 5.7, students encounter the relationship between zeros and factors of a polynomial. The teacher’s materials present this information, but students do not develop understanding of the relationship by investigating or explaining the relationship on their own.
• A-REI.4.10: In Algebra 1, Lesson 2A.2, the materials state that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). However, students do not develop understanding of this standard, as they substitute numbers into equations and plot points to produce the graphs.
• F-LE.1.1: In Algebra 1, Lesson 2A.7, students interpret key features from linear and exponential graphs and sketch graphs from a verbal description. In lesson 2A.9, students calculate average rate of change of a function, and in Lesson 2A.14, students create exponential functions from context, determine the rate of change, and compare exponential functions to linear functions. In Algebra 2, Lesson 3B.12, students select a function to represent real-world problems, analyze data sets, identify the domain of the data sets, and the rate at which the range is changing in order to select a model that best fits the data set. Students’ conceptual understanding of exponential and linear functions is reduced in these lessons due to the amount of scaffolding and step-by-step instructions provided for the students.
• G-SRT.3.6: In Geometry, Lesson 2.10, students use ratios and proportions to determine missing lengths. In the Scaffolded Practice, students examine corresponding angles and sides of similar triangles, but in the Practice, students do not explain how the side ratios in right triangles lead to the definitions of trigonometric ratios. The teacher materials provide scaffolded questions leading students to the connections between sides of similar triangles and the definition of trigonometric ratios, but students do not independently demonstrate conceptual understanding of this standard.

The instructional materials for the Walch Traditional Florida series partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Each lesson is the same throughout the series beginning with a Warm-up, Scaffolded and Guided Practice, which typically address procedural skills, and a Problem-Based Task. The materials incorporate conceptual understanding in Extending the Task.

The three aspects are not balanced with respect to standards being addressed as minimal evidence of conceptual understanding can be found throughout the materials. The majority of lessons address procedural skills and application. These two aspects of rigor are sometimes presented separately and sometimes presented together in the same lesson. The instructional materials balance procedural skills with application throughout the series, and procedural skills are practiced within the context of application problems.

Examples of how conceptual understanding is not presented together with either of the other aspects of rigor include:

• In Algebra 1, Lesson 1.6, students create and graph linear equations in two variables (A-CED.1.2, N-Q.1.1). Throughout the Scaffolded Practice and Guided Practice, students create a linear equation from a real world example, such as hourly rates for a job. From student-generated graphs, students identify the slope and y-intercept and give meaning to those values in context. In the Problem-Based Task, students apply linear equations to purchasing phone cards. Students create a coordinate plane, determine the scale, write the linear equation, and interpret the meaning of their solutions. Application and procedural skills are addressed together.
• In Geometry, Lesson 1A.4, students define specific reflections and rotations as well as the general form for a translation, and students describe and manipulate functions algebraically (G-CO.1.4). In the Problem-Based Task, students analyze a series of transformations performed on a surfboard as it goes through a conveyor belt and describe an equation to represent the transformation. Application and procedural skills are addressed together.
• In Algebra 2, Lesson 2.2, students examine radians and how radians relate to the unit circle (F-TF.1.2). Throughout the Guided Practice and Scaffolded Practice, students sketch radian measurements on the unit circle. In Example 3 of the Guided Practice, students use the coordinate plane to demonstrate why the point where the terminal side intersects the unit circle is (cos x, sin x), but there are no opportunities for students to perform this independently. The Problem-Based Task relates the unit circle to a bicycle tire. A sticker is on the tire and rotates as the wheel moves counterclockwise. Students determine how far the sticker is from the ground after a given amount of time utilizing angle measurement in radians, arc length, and the radius of the tire. Application and procedural skills are addressed together.

The instructional materials for the Walch Traditional Florida series partially meet expectations for supporting the intentional development of reasoning and explaining (MP.2.1 and MP.3.1), in connection to the high school content standards. The materials develop MP.2.1 to its full intent, but the materials do not develop MP.3.1 to its full intent. There are several Problem-Based Tasks where students explain their reasoning or construct an argument, but no evidence was found where students critique the reasoning of others. The implementation guides for the Problem-Based Tasks often encourage teachers to have students share their thinking, but they do not include prompts for students to critique others’ thinking.

Examples of the materials supporting the intentional development of MP.2.1 include:

• In Algebra 1, Lesson 2B.6, Problem-Based Task, students use detailed information about the sale of 2-day and 3-day tickets for adults and children. Students reason about the quantities provided as they define variables to represent different quantities and use those variables to write a system of equations describing the constraints of the task.
• In Geometry, Lesson 2.2, Problem-Based Task, students determine how to enlarge a picture from 5x7 to 8x10 without distorting the picture. Students reason about the physical scenario of enlarging a picture and make connections between that and the abstract mathematical work of performing a dilation.
• In Algebra 2, Lesson 3A.7, Problem-Based Task, students reason abstractly by determining how to organize data presented to them in a paragraph so they can find a logarithmic function that models the data. Students also reason quantitatively by determining if the corresponding exponential function models the given data.

Examples of the materials not supporting the intentional development of MP.3.1include:

• In Algebra 1, Lesson 2B.8, Problem-Based Task, students write and graph an inequality representing the number of friends who can be assigned to help make cupcakes if there are at most 5 friends available. The Implementation Guide has teachers encourage students to work together to discuss their methods. Students construct viable arguments determining whether the inequality is inclusive or non-inclusive. Students choose test points to determine if the solution of the inequality allows for viable arguments and whether any of the student solutions are correct. Students do not critique the reasoning of others.
• In Geometry, Lesson 1B.2, students prove or disprove a statement made about a stained glass art pattern. Students construct a viable argument, but they are not prompted to critique the reasoning of others.
• In Algebra 2, Lesson 1A.6, Problem-Based Task, students solve a quadratic equation using the quadratic formula, property of square roots, and factoring. The Implementation Guide states, “When presenting their arguments to others, students will explain their results and validate that each method resulted in the same solutions, as they illustrate how each method produced solutions that can be displayed in the same formats.” There are no directions that involve students analyzing the arguments of others.

The instructional materials for the Walch Traditional Florida series partially meet expectations for supporting the intentional development of modeling and using tools (MP.4.1 and MP.5.1), in connection to the high school content standards. The materials develop MP.4.1 to its full intent, but the materials do not develop MP.5.1 to its full intent.

Examples of the materials supporting the intentional development of MP.4.1 include:

• In Algebra 1, Lesson 5.16, Problem-Based Task, students help Fatima determine which is the better of two options for buying a car. Students formulate a mathematical model to represent each option, and students perform calculations for each model. Students interpret their results and report their recommendation on which option to select.
• In Geometry, Lesson 3.5, Problem-Based Task, students are presented with a table of data, and a scatterplot of the data, relating the density of melting ice to its temperature. Students examine the table and plot to determine what type of function might best model the data. After picking a type of function, students can interpret how well their selection models the data and make revisions as needed.
• In Algebra 2, Lesson 1B.9, Problem-Based Task, students create a graph and make a recommendation to a company about the size of cans for a new product. Students engage in the full intent of MP.4.1 as they formulate possible solutions, compute, interpret, and validate their findings.

Examples of the materials not supporting the intentional development of MP.5.1include:

• In Algebra 1, Unit 2A, Station Activities, students are directed to use graphs, rulers, calculators, and tables for comparing linear models, but students do not choose which tools to use.
• In Algebra 1, Lesson 5.7, Problem-Based Task, students analyze profits modeled by a polynomial function and determine when the company shows a zero profit, positive profit, and a loss. Students are directed to sketch a graph.
• In Geometry, Lesson 3.5, Problem-Based Task, students are given a graph and a table showing the density of water at different temperatures. Students are directed to use a graphing calculator to find an equation relating temperature to density. Students do not choose the appropriate tool.
• In Algebra 2, Lesson 1A.16, students use the equation representing the volume of a hot-water tank to determine if there is enough space for the hot-water tank to fit in the space allotted in a basement. The coaching questions direct students to utilize a graphing calculator as the tool, rather than having students choose a tool. Students are also directed to use synthetic division and do not choose a method of their own for solving the problem.

The instructional materials for the Walch Traditional Florida series partially meet expectations for supporting the intentional development of seeing structure and generalizing (MP.7.1 and MP.8.1), in connection to the high school content standards. The materials develop MP.7.1 to its full intent, but in the few instances where MP.8.1 was identified in the materials, the tasks did not align to MP.8.1.

Examples of the materials supporting the intentional development of MP.7.1 include:

• In Algebra 1, Lesson 2A.8, Problem-Based Task, students analyze a table of data that represents two methods an insurance company uses to calculate the worth of a desk over a period of time. Students apply the structure for rate of change to the two scenarios as they calculate rates of change over an interval. The structure of rate of change is also applied to a nonlinear function.
• In Geometry, Lesson 3.2, Problem-Based Task, students look for and make use of structure as they determine how many square feet on the ground will be saved when three conical piles of sand are moved to one cylindrical storage tank.
• In Algebra 2, Lesson 1A.20, Problem-Based Task, Luca pours water into a 5-gallon bucket. He initially pours 200 ounces into the bucket, and on each subsequent pour, he adds half of the previous amount. Students make use of structure to determine if Luca will add so much water that it overflows the bucket.

Examples of the materials not supporting the intentional development of MP.8.1 include:

• In Algebra 1, Lesson 1.11, Problem-Based Task, students determine how many years until the number of households with only cell phone service would be greater than the number of households with both cell phones and landlines. According to the Implementation Guide, MP.8.1 is addressed as, “encourage students to draw parallels between and make generalizations about solving an equation compared to solving an inequality.” Students do not express regularity in repeated reasoning in this task.
• In Geometry, Lesson 5.1, Problem-Based Task, students verify by measuring and comparing the circumference and diameter of circular objects in the classroom. Students use the same process each time, but they do not look for repeated reasoning nor do they need to express any regularity they find.
• In Algebra 2, Lesson 1A.8, Problem-Based Task, students are given numerical dimensions of a garden, and within a diagram of the garden, several sections are represented with variable expressions. Students determine the dimensions of one of the sections. The Implementation Guide states MP.8.1 is addressed because students use repeated reasoning as they determine the expressions for the length and width of the corn section. Students add and subtract polynomials, and there is no evidence of looking for or expressing regularity in repeated reasoning.