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Focus & Coherence
Gateway 1 - Meets Expectations | 88% |
|---|---|
Criterion 1.1: Focus & Coherence | 16 / 18 |
Criterion 1.1: Focus & Coherence
Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional series meet expectations for focus and coherence. The instructional materials attend to the full intent of the mathematical content contained in the high school standards for all students, spend the majority of time on the CCSSM widely applicable as prerequisites, let students fully learn almost all non-plus standards, engage students in mathematics at a level of sophistication appropriate to high school, and make meaningful connections in a single course and throughout the series. The instructional materials partially attend to the full intent of the modeling process and partially identify and build on knowledge from Grades 6-8.
Indicator 1a
The materials focus on the high school standards.*
Indicator 1a.i
The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional 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 non-plus standards are not addressed across the courses of the series.
The following are examples of standards addressed by the courses of the series:
- N-RN.3: In Algebra I, Module 3, Topic 1, Activity 2.5, students explain how the properties of rational exponents extend from the properties of integer exponents. Students consider calculations between numbers and determine to which number set the answer belongs. For example, students determine if the product of a nonzero rational number and an irrational number is sometimes, always, or never equal to a rational number.
- A-REI.1: In Algebra I, Module 2, Topics 1 and 2, students use properties of equality to solve equations and inequalities. Students check and justify their solutions.
- F-IF.7c: In Algebra II, Module 1, Topic 2, Activity 1.1, students identify zeros, predict the shape of a function by sketching the graph and labeling the zeros, and check their solutions using a graphing calculator. Students use technology to graph, identify domain and range, determine intervals of increasing and decreasing, and the x- and y-intercepts. In the same module, Topic 3, Activity 3.2, students encounter the definition of end behavior and complete a table to describe end behavior.
- G-CO.3: In Geometry, Module 1, Topic 3, Lesson 5, students describe symmetry in a rectangle and reflectional symmetry in a square. Students also determine if shapes have rotational or reflectional symmetry and are asked to “describe the reflections and rotations that can carry each figure onto itself.”
- S-CP.3: In Geometry, Module 5, Topic 2, students use conditional probability to solve problems and identify independent or dependent events. Also in the topic, students use an example of conditional probability and determine if the probability of A given B is the same as the probability of A.
The following non-plus standards are not fully addressed:
- A-REI.5: There was no evidence found where the materials or students prove, or are shown how to prove, that given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Indicator 1a.ii
The materials attend to the full intent of the modeling process when applied to the modeling standards.
The instructional materials reviewed for Carnegie Learning Math Solutions Traditional series partially meets expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials do not use the full intent of the modeling process to address more than a few modeling standards across the courses of the series.
Examples where the full modeling process was present include:
- G-SRT.1: In Geometry, Module 3, Topic 2, Assessments, Topic Performance Task, Trigonometry: “It’s a Bird! It’s a Plane! It’s … a Drone?”, students are given information of drones flown by Jeremy and Leslie. Students determine which drone is higher, the difference between the heights, and the distance between Jeremy’s and Leslie’s locations on the ground. Students consider if Jeremy flies his drone the same height as Leslie’s drone, and they explain why the angle of elevation from Jeremy’s location is different from the angle of elevation between Leslie’s location on the ground and her drone. Students calculate the angle of elevation from Jeremy’s location on the ground to his drone to validate their reasoning. According to the rubric, students provide a labeled drawing, including all given measurements and calculated measurements. Students formulate their own equation in order to compute whose drone is higher and the difference between the heights of the drone. Students compute the distance between Jeremy and Leslie with supporting work. Students provide an explanation as well as validation to support their work.
- N-Q.2 and S-ID.6a: In Algebra I, Module 3, Topic 2, Lesson 4, students work with probabilities of real-world scenarios. Students complete a series of analysis questions and draw conclusions from the probabilities. In Activity 4.1, students make predictions, create a model with a table, graph, and equation, and define the variables. In Activity 4.2, students summarize the results and write out the work they did in each step of the modeling process while completing Activities 4.1 and 4.2. Students choose functions and utilize statistical skills by creating charts or tables to represent data. Students decide if the function they wrote is a good fit, and they make predictions and explain answers. Students also write an article for the newsletter for SADD.
All aspects of the modeling cycle are addressed throughout the series, however, students are often limited in their opportunities to make choices and assumptions when defining mathematical modeling problems as well as validating their conclusions and improving upon their models when appropriate. Examples of how students engage in some, but not all, elements of the modeling process include:
- A-SSE.1: In Algebra II, Module 1, Topic 1, Lesson 4, students explore that a distinct line that passes through two points and 3 non-linear points establishes a distinct parabola. Then students compare the forms of parabolas (general, vertex, factored) and write equations from a vertex and point. Next, they write quadratic equations from a specific graph, so only one equation can be written. In the Extension, students solve a problem about laws governing a pool and a diving board. Students explain their reasoning, but there is no evidence that students test solutions, communicate with others, and revise thinking. Students define the variables in the problem.
- F-IF.4: In Algebra II, Module 1, Topic 2, Lesson 2, students are presented 3 patterns in different contexts. Students answer open-ended questions about the growth patterns and how the quantities change relative to one another. Students extend their thinking and write an expression to determine the next tile. Students choose variables but are encouraged to rethink or revise their expressions. Finally, they are given sample student expressions with descriptions and choose the ones that match and explain their reasoning.
- F-BF.1: In Algebra II, Module 2, Topic 2, Lesson 3, Activity 3.1, students describe patterns in a table with two quantities and predict a model that would represent a relationship between the quantities. Students used a blank coordinate graph to draw a scatter plot and technology to write a regression line. Students compare their prediction, graph, and regression equation obtained from technology. Students do not rethink or revise. Aspects of the modeling process included were: make a prediction, analyze predictions, and make choices.
- G-MG.1: In Geometry, Module 4, Topic 1, Lesson 4, students imagine stacking pancakes (circular, square, triangular and decreasing sizes), and students name the shape formed, relating the dimension of a single pancake of that shape to the entire stack and drawing conclusions about the 3D solid created. Students make a pyramid with cubes, explore volume by making changes in dimensions, and build a spreadsheet with the data from the pyramid explored. Students compare their data and write a formula for volume of a pyramid. Students build physical models of 3D shapes, gather data, and make conjectures about volume. Several components of the modeling process are found in this lesson, students ask questions and make assumptions to define the problem, use mathematical tools to solve, and explain if/when their answer makes sense. Students do not test and revise models.
In the materials, many lessons are structured with learning opportunities which contain step-by-step instructions for students with minimal opportunities for creativity, estimation, and student choice of math concepts and skills to combine and utilize for problem solving. At the end of topics and/or lessons, Performance Tasks can be found in assessment sections. The full modeling process is present within these tasks however, the tasks are found within a summary assignment of scaffolded lessons which directs how students should mathematize the problem along with predictions and analyses that should occur.
Indicator 1b
The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
Indicator 1b.i
The materials, when used as designed, allow 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.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional series, when used as designed, meet expectations for allowing students to spend the majority of their time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers. The materials do not include distracting content that would keep students from engaging with the standards identified as widely applicable prerequisites (WAPs).
Examples of how the materials allow students to spend the majority of their time on the WAPs include, but are not limited to:
- In Algebra I, Module 3, Topic 2, Lesson 2, students rewrite radical expressions and rational expressions using the properties of exponents. In Geometry, Module 1, Topic 1, Lesson 4, students rewrite radical expressions arising from the distance formula and Pythagorean Theorem. In Geometry, Module 2, Topic 2, Lesson 4, students use and rewrite rational expressions arising from trigonometric ratios in special right triangles. In Algebra II, Module 3, Topic 1, Lesson 4, students write radical expressions with rational exponents, rewrite the radicands of variable expressions, and compute the sums/differences of irrational numbers in algebraic expressions. (N-RN)
- In Algebra 1, Module 5, Topic 1, Activity 2.4, students factor quadratic expressions and graph them in order to find the zeros (A-SSE.3a). Also, in Algebra 1, Module 1, Topic 2, Lesson 4, students complete the square to transform quadratics into vertex form and find zeros in order to graph quadratics (A-SSE.3b).
- In Algebra I in the MATHia software, students explore constant change, evaluate linear functions, complete charts, identify the input value, compute the equation by substituting, and determine the output value. In Algebra II, Module 1, Topic 3, students explore the characteristics of polynomial functions. In Algebra II, Module 3, Topic 4, students write equations of exponential functions from patterns. In Algebra II, Module 4, students explore trigonometric functions. (F-IF)
- In Geometry, Module 3, Topic 1, students use dilations to create similar figures and establish criteria for determining similar triangles. Students use similarity to establish proportionality theorems and use similar triangles to solve problems (G-SRT.A, G-SRT.B). In Geometry, Module 3, Topic 2, students use similarity to determine constant ratios in right triangles and define trigonometric ratios (G-SRT.C).
- The Statistics WAPs are addressed in Algebra I and Algebra II, and students have multiple opportunities to engage with the WAPs from this category.
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional series, when used as designed, meet expectations for letting students fully learn each non-plus standard. However, the instructional materials for the series, when used as designed, do not enable students to fully learn a few of the non-plus standards.
Examples of the non-plus standards that would not be fully learned by students when using the materials as intended include:
- A-SSE.4: In Algebra 2, Module 3, Topic 4, Activity 1.1, students do not derive the formula for a geometric series. An example is provided and students analyze the example to find a pattern in one question with two parts. Underneath the question, the materials give the formula to compute any geometric series. Students use the geometric series to solve problems.
- A-REI.4a: In Algebra 1, Module 5, Topic 2, Lesson 5, students complete the square in order to solve quadratic equations. The materials derive the quadratic formula by completing the square, but students do not derive the quadratic formula on their own.
- A-REI.11: Students have limited opportunities to explain why the x-coordinates of the points where the graphs of two equations intersect are solutions. In Algebra 1, Module 2, Topic 3, Lesson 1, students find the intersection of two linear equations and explain why the x- and y-coordinates of the points where the graphs of a system intersect are solutions. In Algebra 1, Module 5, Topic 1, Activity 1.3, students find the intersection of constant and quadratic equations and explain why the x- and y-coordinates of the points where the graphs intersect are solutions. In Algebra 1, Module 5, Topic 3, Lesson 2, students find the solutions to systems of quadratic equations. Students do not explain this relationship for absolute value, rational, exponential, and logarithmic functions.
- G-C.5: In Geometry, Module 4, Topic 1, Lesson 2, Getting Started, students use a dartboard of 20 sectors to determine the area of the entire dartboard and the area of one sector. Then students find the area of one sector if the dartboard was divided into 40 sectors. Students do not generalize their findings to a dartboard with n sectors and are given the formula for the area of a sector at the beginning of Activity 2.1.
Indicator 1c
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional 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 of applying key takeaways include, but are not limited to:
- In Algebra I, Module 1, Topic 1, students interpret key features of graphs in the context of a problem (8.F.4). In Activity 1.1, students read scenarios, determine the graph which represents the scenario, and identify independent and dependent quantities. In Activity 1.2, students compare and contrast the graphs they organized in the first sections.
- In Geometry, Module 3, Topic 1, students develop similarity standards (8.G.4). In Activity 1.1, students dilate figures to create similar figures. In Activity 1.2, students establish similarity criteria. In Activity 5.1, students apply knowledge of similar triangles in order to solve mathematical problems.
- In Algebra I, Module 5, Topic 2, Lesson 5, students solve problems and apply knowledge of real numbers to determine to which set the answers belong. In Algebra II, Module 1, Topic 1, Lesson 6, students apply knowledge of real numbers to determine if solutions are a part of the real number system. These activities apply key takeaways from 6-8.NS involving integers, rational numbers, and irrational numbers.
Examples of age appropriate contexts throughout the series include, but are not limited to:
- In Algebra I , Module 1, Topic 3, Activity 3.1, students construct a scatter plot that relates speed and braking distance.
- In Geometry, Module 3, Topic 2, Lesson 2, Talk the Talk task, students relate road grades of mountainous areas to the angle of elevation.
- In Algebra II, Module 1, Topic 1, Activity 1.2, students use a scenario regarding homecoming elections and how the results have been shared. Also in this lesson, students explore the pattern of a rumor spreading.
- In Algebra II, Module 1, Topic 1, Activity 5.3, students apply a quadratic model to launching a free t-shirt to fans at a baseball game.
The instructional materials use various types of numbers throughout the series in expected places where numbers are naturally varied, such as trigonometry, regression models, and problems involving radicals and/or rational exponents. Examples of the materials using various types of numbers include, but are not limited to:
- In Algebra I, Module 5, Topic 2, Activity 5.4, students evaluate the discriminant of quadratic equations and the calculations include real, rational, irrational, and imaginary numbers.
- In Geometry, Module 4, Topic 2, Lesson 3, students determine the intersection points for each system of equations involving a circle, including rational and irrational coordinates.
- In Algebra II, Module 4, Topic 2, students use integers, rational numbers, and irrational numbers in trigonometric functions and relationships.
Indicator 1d
The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional 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 of connections made across courses include, but are not limited to:
- In Algebra I, Module 3, Topic 1, and Algebra II, Module 3, Topic 1, students use rational exponents and their definitions in order to solve problems (N-RN.1).
- In Algebra I, Module 5, Topics 1and 2, and Algebra II, Module 2, Topic 1, factoring, completing the square, and finding equivalent forms are connected. Students write equivalent forms and make connections between solutions, graphs, and other key details using the equivalent forms (i.e., connections between factors and zeros and completing the square to reveal the vertex or as an alternate way of solving an equation) (A-SSE.3).
- In Algebra I, Topic 1 of Modules 1, 2, 3, and 5, and Algebra II, Module 2, Topic 1, Module 3, Topics 1-3, and Module 4, students use functions to model different situations. Although often confined to a specific type of function, students complete a process to develop understanding of why a certain function type is used in a context (F-IF.4).
- In Algebra II, Module 1, Topic 3, and Module 2, Topic 3, Lesson 3, students transform rational and polynomial functions and sketch a graph involving a transformed function. Students apply knowledge of geometric transformations from Geometry Module 1 to transform a graph based on an equation (G-CO).
- In Geometry, Module 4, Topic 2, Lesson 2, students use completing the square as a method for writing the equation of a circle. Students also use completing the square as a way to calculate minimum and maximum values in Algebra I, Module 5, Topics 6 and 7, and in Geometry, they determine the equation of a circle in standard form using the same method (A-SSE.B).
The following examples are instances where meaningful connections are made within courses:
- In Algebra I, average rate of change is addressed through linear functions in Module 2 and students explore average rate of change with exponentials in Module 3, including connections to a constant ratio for geometric sequences and the use of MATHia Software. Average rate of change is explored for quadratic functions in Module 5 (F-IF.6).
- Transformations are used throughout the Geometry course. In Module 1, Topic 1, students are introduced to the idea of transformations where definitions for both transformations and rigid motion are given. In Topic 3, students explore the different rigid motions using simple geometric transformation machines, similar to function machines. In Module 2, Topic 1, Lesson 2, students use transformations to prove triangle congruence, and in Lesson 3 students use triangle congruence to solve problems (G-CO).
- In Algebra II, students connect patterns to functions, tables, and graphs. In Module 1, students match scenarios, various function forms, tables, and graphs with linear, exponential, and quadratic functions. In Module 2, students factor, use long division, sketch graphs, and analyze tables. Students also write equations to solve problems from scenarios (A-SSE).
Indicator 1e
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards.
The Teacher Implementation Guide includes “What is the entry point for students” in each topic overview. The instructional materials do not explicitly identify content from Grades 6-8. The materials reference middle school learning but not by specific standards. The materials support the progression of the high school standards by including connections between content from Grades 6-8 and high school that enable students to extend their previous knowledge. Examples from teacher materials include but are not limited to:
- In Algebra I, Module 2, Topic 2 Overview, “In grade 7, students solved two-step inequalities and graphed the solutions on a number line. Students build from this knowledge in this topic when they solve more complex inequalities in one variable and compound inequalities.”
- In Geometry, Module 1, Topic 3 Overview, “Students know that two figures are congruent if and only if there exists a sequence of one or more rigid motions that carries one of the figures onto the other.”
- In Algebra II, Module 5, Topic 1 Overview, “Since middle school, students have created and analyzed data in a variety of distributions, and have compared different displays.” “This topic deepens students’ understanding of the importance and usefulness of these measures and provides them with an opportunity to apply these concepts in real-world situations.”
The Student Editions of the instructional materials include a family guide with “Where have we been? Where are we going?” for each topic. These sections identify connections between middle school content and courses and lessons in the series. The materials refer to middle school but not the standard(s) being built upon throughout the lesson. Examples include but are not limited to:
- In Geometry, Module 3, Topic 1, “In middle school, students developed their understanding of proportional reasoning through explorations of multiplicative relationships. Students have used scale factor to solve problems.” “Understanding similarity further develops proportional reasoning which began in grade 6 and continues throughout high school mathematics. It provides the opportunity for students to connect spatial and numeric reasoning and lays the groundwork for understanding trigonometric ratios, which students will explore in the next topic.”
- In Algebra I, Module 2 ,Topic 3, “Coming into this topic, students know that every point on the graph of an equation represents a value that makes the equation true.” “Students have written systems of linear equations and have solved them either graphically or algebraically using substitution.”
Tables are presented as an overview of each module in the teacher editions and contain a column for connections to prior learning. This is evident for each module, but explicit standards are not listed in this overview or anywhere else in the teacher and/or student materials.
Indicator 1f
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.