High School - Gateway 2
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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 68% |
|---|---|
Criterion 2.1: Rigor | 6 / 8 |
Criterion 2.2: Math Practices | 5 / 8 |
Criterion 2.1: Rigor
Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.
The instructional materials for the Walch Traditional Florida series partially meet expectations for reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations. The instructional materials meet expectations for developing procedural skills and providing opportunities to engage in applications. The instructional materials partially meet expectations for developing students’ conceptual understanding and ensuring that the three aspects of rigor are not always treated together and are not always treated separately.
Indicator 2a
Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.
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.
Indicator 2b
Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.
The instructional materials for the Walch Traditional Florida series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills throughout the series. Opportunities for students to independently demonstrate procedural skills across the series are included in each lesson through Scaffolded Practice and Guided Practice. The Scaffolded Practice is a set of 10 practice problems providing practice during instructional time. The Guided Practice examples start with step-by step prompts for solving and finish with Guided Practice examples without prompts.
Examples that show the development of procedural skills and opportunities for students to independently demonstrate procedural skills across the series include:
- A-SEE.1.2: In Algebra 1, Lessons 4.7-9, students identify the structures of quadratics based on the leading coefficients and rewrite them in equivalent factored forms. In Algebra 2, Lessons 1A.10-12, students use the structure of an expression to identify ways to rewrite it through polynomial identities, complex polynomial identities, and the binomial theorem.
- A-APR.1.1: In Algebra 1, Lessons 4.4 and 4.5, students find sums, differences, and products of polynomial expressions. In Algebra 2, Lesson 1A.8, students add and subtract polynomial expressions, and in Lesson 1A.9, students multiply polynomial expressions.
- A-APR.4.6: In Algebra 2, Lesson 1B, there are four lessons in which the materials develop procedural skills with operations on rational expressions, and students independently demonstrate those procedural skills.
- F-BF.2.3: In Algebra 1, Lesson 2B.4, students graph multiple functions, compare the functions, and describe the functions in terms of geometric translations of another function. In Algebra 2, Lessons 5.17 and 5.18, students practice transforming quadratic functions.
- G-GPE.2.7: In Geometry, Lesson 4.4, Scaffolded Practice, students compute perimeter and area given the vertices of quadrilaterals and triangles. In Guided Practice, students use the distance formula to determine the perimeter of the polygons. Throughout the practice worksheets, students compute the area and perimeter of polygons using coordinates.
Indicator 2c
Attention to Applications: The materials support 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.
The instructional materials for the Walch Traditional Florida series meet expectations for supporting 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. The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematics throughout the series. The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts through the Problem-Based Tasks located in each lesson.
Examples from the instructional materials that demonstrate opportunities for students to engage in routine and non-routine application of mathematics throughout the series include:
- A-SSE.2.3: In Algebra 2, Lesson 1A.4, Pyroclastic Projectiles, students determine how long they expect a volcanic bomb to remain in the air if it was ejected from a volcanic eruption with a given velocity and height. Students use multiple methods to solve the problem, decide which is solution is best, and justify the answer.
- A-REI.4.11: In Algebra 2, Lesson 1B.10, Connectivity Calculations, students determine how far away from the front wall of a house the WiFi can be accessed while on the sidewalk that passes in front of the house.
- F-IF.2: In Algebra 2, Lesson 3B.12, Comparing Social Media Growth, students determine if exponential, linear, or quadratic functions best model the data points that represent the growth of users of a social media website.
- G-SRT.3.8: In Geometry, Lesson 2.12, students apply the Pythagorean Theorem to validate the diagonal measurement of a television set. In Guided Practice, students calculate the dimensions of a courtyard using trigonometric ratios. In the Practice Student Worksheets, students use trigonometric ratios to determine how far up the side of a building a ladder will reach.
Indicator 2d
Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.
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.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for the Walch Traditional Florida series partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials meet expectations for the intentional development of overarching, mathematical practices (MP.1.1 and MP.6.1), and the instructional materials partially meet expectations for the remainder of the indicators in this criterion.
Indicator 2e
The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.
The instructional materials for the Walch Traditional Florida series meet expectations for supporting the intentional development of overarching, mathematical practices (MP.1.1 and MP.6.1), in connection to the high school content standards. In the Program Overview of the Teacher’s Edition, a Correspondence to Standards for Mathematical Practice is included. Within each lesson, a Problem-Based Task denotes the Standards for Mathematical Practice specific to that task. The majority of the time MP.1.1 and MP.6.1 are used to enrich the mathematical content. Across the series, there is intentional development of MP.1.1 and MP.6.1 that reaches the full intent of the Mathematical Practices.
Examples of the materials supporting the intentional development of MP.1.1 and MP.6.1 include:
- In Algebra I, Lesson 2A.7, Problem-Based Task, students consider the year, make, model, included options, and price of new cars. Students consider how the car will decrease in value over time. The teacher’s guide has Facilitating the Task, which discusses student behavior that will demonstrate the mathematical practices. For this Problem-Based Task, Facilitating the Task indicates for MP.1.1, “Student will recognize the two quantities that are involved, the quantities will represent an exponential function, and interpret the information in terms of the key features of the graph.” For MP.6.1, “correct terms and notation for domain, asymptote, and maximum. Look for proper units of years and dollars.”
- In Algebra 1, Lesson 4.1, Problem-Based Task, students determine the terms, factors, and coefficients of a quadratic expression to represent the area of a deck to be decorated. Students make sense of the shape of the deck and clarify the meaning of altitude, coefficient, factor, quadratic expression, and term. In the Problem-Based Implementation Guide, teachers are given suggestions to reinforce MP.1.1 for their students. Students persevere as they work with order of operations, translating verbal descriptions to algebraic expressions, and like terms.
- In Algebra 1, Lesson 2A.5, students work with function notation and evaluating functions. Students are precise with function notation, f(x), and the language that the range of a function is dependent on its domain. Students identify the domain and range in set notation, and use function notation in their work with functions, exemplifying MP.6.1.
- In Geometry, Lesson 1A.3, Problem-Based Task, students examine a plan for a park that must meet certain requirements regarding symmetry. Students determine where trees must be planted in order to meet these requirements. Students make sense of the stated conditions and requirements (MP.1.1) in order to plant the trees in appropriate places. Students also attend to precision by considering lines of symmetry and using the coordinate grid to precisely mark the location of the trees (MP.6.1).
- In Geometry, Lesson 1A.6, students describe how to navigate an unmanned vessel through the ocean floor to recover an artifact from the Titanic. Students specify the transformations that should be used to avoid obstacles and retrieve the artifact. Students use precision in their descriptions of the transformations (MP.6.1).
Indicator 2f
The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.
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.
Indicator 2g
The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.
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.
Indicator 2h
The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.
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.