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Focus & Coherence
Gateway 1 - Meets Expectations | 83% |
|---|---|
Criterion 1.1: Focus & Coherence | 15 / 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 Mathematics Vision Project Traditional series meet the expectation for focusing on the non-plus standards of the CCSSM. The Modules and Tasks across the series are organized in a consistent logical structure of mathematics. Overall, the instructional materials attend to the full intent of the non-plus standards, attend to the full intent of the modeling process, spend a majority of time on the widely applicable prerequisites from the CCSSM, require students to engage at a level of sophistication appropriate to high school, and make meaningful connections within each course and throughout the series.
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 Mathematics Vision Project Traditional series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. There are some standards for which the instructional materials attend to parts of the standard and some standards for which the instructional materials do not attend to the standard.
The following are examples for which the materials attend to the full intent of the standard:
- A-SSE.3: In Algebra II, Module 3, Task 7, students verify different forms of a quadratic expression to solve a given equation. Students explain how the factored form helps to reveal the zeros and what that means in the context of the Curbside Rivalry question. In Algebra I, Module 2, Task 6, students are guided through an exploration of how expressions with different rational exponents are equivalent, yet highlight different mathematical properties.
- G-MG.1: In Geometry, Module 7, Ready, Set, Go! Problem 3, students use a model to find the total surface area and volume of the Washington Monument. In Geometry, Module 7, Task 4, students model how they would determine the volume of a nail.
- S-IC.2: In Algebra II, Module 9, Task 7, students analyze a model created by a “slacker” student for a true/false quiz. Within this task, students complete an analysis of his model and, at the same time, test their analysis using coin flips.
The materials attend to some aspects, but not all, of the following standards:
- F-IF.6: In Algebra I, Module 8, Task 2, students calculate the average rate of change from piecewise functions. In the majority of the examples, students calculate a constant rate of change from linear, piecewise functions. The materials do not include estimating the rate of change from a graph, only equations or functions.
- F-IF.7b: Cube root functions or graphs are not present in the materials.
- G-CO.1: The definitions are present within Geometry Module 1, Task 4, but the definitions are not based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
- N-Q.3: Students do not choose a level of accuracy. In Geometry, Module 7, Task 8, students are directed to round to the nearest centimeter, but students do not choose the level of accuracy for themselves.
The following standards are not attended to by the materials:
- A-SSE.4: The materials do not have a derivation of the formula for the sum of a finite geometric series. This standard was not identified in the materials.
- S-IC.4: Students do not use data from a sample survey to estimate a population mean or develop margins of error. This standard was not identified in the materials.
- S-IC.5: There is a discussion of how students could randomly assign participants in an experiment in Algebra II, Module 9, Task 5, but there is no use of simulations. This standard was not identified in the materials.
- S-IC.6: Students do not evaluate reports based on data. This standard was not identified in the materials.
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 Mathematics Vision Project Traditional series meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. The materials provide opportunities for students to engage in the modeling process. Tasks that involve modeling include a graphic of the modeling process in the teacher notes. Additionally, the modeling standards are addressed in the materials.
Examples of modeling tasks include:
- In Algebra I, Module 8, Task 3, students interpret a graph detailing Michelle’s bike ride to and from a lake. Students are asked to create a function to model the situation.
- In Algebra I, Module 3, Task 1, students sketch a graph given steps that Sylvia used to clean and refill her pool (F-IF.4). Students answer provided questions to complete the problem.
- In Geometry, Module 4, Tasks 10 and 11, students complete real-world problems with angles of elevation, angles of depression, and right angles (G-SRT.8).
- In Geometry, Module 5, Task 11 is designed to “deepen their understanding of volume formulas” (G-GMD.3). Students discuss why the formula for the volume of a cone is one-third the volume of a prism. Students compare the two volumes.
- In Algebra II, Module 1, Task 2 details mathematical modeling completed by police departments and insurance companies to determine how far a car goes once it begins to break (in order to solidify the topic of an inverse function, F-BF.1).
- In Algebra II, Module 9, Task 2, students analyze and determine a “good” score on the ACT given information about the mean and standard deviation (S-ID.1). Students answer analysis questions that are provided.
While there are many examples of modeling problems throughout these materials, there are some problems labeled as “modeling” problems that provide scaffolding which inhibits students from engaging in the full modeling process.
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 Mathematics Vision Project Traditional series meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs and careers. There was a large focus on WAPs in the Algebra I course with a decreasing amount of tasks in the subsequent courses. Throughout all three courses, the majority of the tasks spend time developing student understanding of the WAPs. Throughout the materials, there are a limited number of times that students spend too much time on prerequisite skills or distracting material.
Within the WAPs, the largest focus was on the Algebra and Function standards. The Geometry WAPs were cited only in the Geometry course. The Number and Quantity and Statistics WAPs were addressed the least by the materials.
The WAPs from Number and Quantity are included in all three courses. Evidence is found in Algebra I, Modules 1, 2, 3, 4, 5 ; Geometry, Module 7; and Algebra II, Module 3.
The WAPs from Functions are included in Algebra I and Algebra II. Evidence is found in Algebra I, Modules 1, 2, 3, 4, 5, 6, 7 and Algebra II, Modules 3, 4, 5.
The WAPs from Algebra are included in Algebra I and Algebra II. Evidence is found in Algebra I, Modules 1, 2, 3, 5, 6, 7, 8 and Algebra II, Modules 1, 2, 3, 4, 5, 6, 7, 8.
The WAPs from Geometry are included in Geometry. Evidence is found in Geometry, Modules 1, 2, 3, 4, 7.
The WAPs from Statistics and Probability are included in Algebra I and Algebra II. Evidence is found in Algebra I, Module 9 and Algebra II, Module 9.
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for the Mathematics Vision Project Traditional series partially meet the expectation for, when used as designed, allowing students to fully learn each standard. The instructional materials address many standards in a way that would allow students to fully learn them. There are some standards, however, that are not fully addressed, or the instructional materials do not provide enough opportunities for students to practice and learn the standards fully.
The following are examples where the instructional materials partially meet the expectations for allowing students to fully learn a standard:
- N-CN.1: Students name the complex conjugate (Algebra II, Module 3, Ready, Set Go 6), but there is not enough practice for students to fully use the complex conjugate.
- N-CN.2: Students use the relation $$i^2=-1$$ and multiply the imaginary parts of complex numbers (Algebra II, Module 3, Ready, Set, Go! Problemm 5), but there is not enough practice for students with the commutative, associative, and distributive properties of complex numbers.
- A-APR.1: Students practice this standard in Algebra II, Module 3, Task 1, and do not practice it after this introductory task. The materials develop how the polynomials are analogous under the operation of division, but they do not develop how they are analogous under the operations of addition, subtraction, and multiplication.
- A-APR.4: In Algebra I, Module 7, Ready, Set, Go! Problem 10, students verify the factored form of a quadratic is the same as standard form. In Geometry, Module 6, Ready, Set, Go! Problem 4, students verify Pythagorean Triples, and in Algebra II, Module 4, students factor sums and differences of cubes. These are all aspects of this standard, but students do not prove the polynomial identities to use them to describe relationships.
- A-APR.6: In Algebra II, Module 5, Ready, Set, Go!, students complete Problems 15-18 and the related task to determine the simplified form of a rational expression. Three of these problems result in an improper rational expression which enables the students to rewrite the expression. This is not enough practice to fully learn this standard.
- F-IF.9: Throughout the materials, students often have to compare two or more representations of functions, but they do not compare different types of functions through different representations.
- F-BF.1b: Students do not combine various types of functions until Algebra II, Module 3, Task 6 when combining trigonometric functions with other functions.
- F-TF.8: Students are not provided opportunities to practice finding the measure of an angle in any quadrant.
- S-ID.1: In Algebra I, Module 9, Ready, Set, Go! Problem 2, students are provided a recommendation to use a dot plot, but there are no mention of dot plots in the instructional materials beforehand. There is not enough practice of all the types of plots for students to fully learn this standard.
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 Mathematics Vision Project 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 where the materials use age appropriate contexts include:
- In Geometry, Module 7, Task 3, students find the volume of a frustum (created by rotating a trapezoid around the y-axis) and approximate the volume of a vase by replacing the curved edges of the vase diagram with segments. Teachers have students share several different strategies for approximating the volume. (G-GMD.1,4)
- In Algebra 2, Module 7, Task 1, students use the information from Ferris Wheel tasks in previous modules to develop strategies for transforming the functions to represent different initial starting positions for the rider. Students focus on horizontal translations and may recognize that either sine or cosine functions can be used with an appropriate horizontal shift. (F-TF.5, F-BF.3)
- In Algebra 2, Module 8, Task 2, students sketch the shape of given graphs and give reasoning for their sketches. These functions combine linear, quadratic, absolute value, and trigonometric functions. While doing this, students design plans for a new amusement park ride. (F-BF.1b)
Examples where the materials use various types of real numbers include:
- In Algebra I, Module 2, Task 6, students verify that the properties of integer exponents also apply to rational exponents. Students use exponent rules to write equivalent forms of expressions involving rational exponents and rational bases. Expressions include rational numbers in the base, as well as in exponents. (N-RN.1,2, A-SSE.3)
- In Algebra 2, Module 3, Task 4, students write the equation of given graphs of parabolas in vertex, standard, and factored forms. Students use irrational numbers and the radical form of i to write the factored form of the equations. Task 5 introduces i, and students write equations for given parabolas using complex and imaginary roots.
Examples where the materials provide opportunities for students to apply key takeaways from grades 6-8 include:
- In Algebra I, Module 3, Task 4, students use a given graph of two functions to answer questions regarding key features of the graph, and students interpret some of the key features. This is an application of a key takeaway from Grades 6-8 in applying basic function concepts to develop/solidify new understanding in this module. (A-APR.1, A-CED.3, A-REI.11, F-IF.7)
- In Geometry, Module 4, Task 1, students are presented a scenario where an employee at a copy center is enlarging a photo for a customer and makes a mistake. Students answer questions to determine what the mistake was and how the employee should have enlarged the photo. Students apply a key takeaway from Grades 6-8 regarding similar figures. (G-SRT.1)
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 Mathematics Vision Project Traditional series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Overall, the materials provide tasks in similar contexts throughout the series, so students can make connections to previous and future learning. The practice problems in Ready, Set, Go! revisit topics in a spiral manner for students to maintain skills throughout the series.
Examples of the instructional materials fostering coherence through meaningful mathematical connections in a single course include:
- In Algebra 1, Module 6, Task 1, students describe a growing pattern which represents a quadratic function. They build upon interpreting expressions and writing recursive and explicit equations from Module 1 (A-SSE.1 and F-BF.1) to develop the idea that quadratic functions show linear rates of change. In Module 6, this is also connected to A-CED.2 as students write equations to represent quadratic relationships.
- In Geometry, Module 4, Tasks 8-11 address trigonometric ratios and using trigonometric ratios to solve right triangles in mathematical and applied problems (G-SRT.6-8). Task 8 builds upon students’ previous understanding of similar triangles to define trigonometric ratios. Tasks 9 and 10 use those understandings to develop relationships between sine and cosine of complementary angles. G-SRT is also connected to F-TF.8 in Task 9 as students justify whether given conjectures are true or false, and three of the questions presented are based on the Pythagorean identity. In Task 11, students solve applied and mathematical problems using all concepts and skills from the previous tasks.
Examples of the instructional materials fostering coherence through meaningful mathematical connections between courses include:
- In Algebra I, Module 1, Task 4, students analyze the pattern of push-ups Scott will include in his workout. Students examine tables, graphs, and recursive and explicit formulas that show how the constant difference is represented in different ways and define the function as an arithmetic sequence. In Algebra II, Module 4, Task 1, students revisit Scott’s workout and develop understanding related to the degree of a polynomial function and the overall rate of change. Students use multiple representations to arrive at this understanding (F-BF.1; F-LE.1-3,5; F-IF.4,5; A-CED.1,2).
- In Geometry, Module 6, Tasks 7 and 8 (G-GPE.2), students define a parabola geometrically using the focus and directrix. In Task 8, students connect this to quadratic functions and parabolas, which were addressed in Algebra I, Modules 6 and 7 (Functions and Algebra conceptual categories). The concepts are further connected in Algebra II, Module 3, Tasks 4 and 5 (A-REI.4, N-RN.3, and N-CN), where students discover a need for complex solutions to quadratic equations and define the imaginary unit.
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 Mathematics Vision Project Traditional series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. In Launch, teachers “review”, “remind”, or “connect” students to work from previous math classes or prior years, but there is no explicit indication of standards from Grades 6-8. Examples where the materials do not explicitly identify standards from Grades 6-8 include, but are not limited to:
- In Algebra I, Module 2, Task 6, students use their understanding of positive whole number exponents to rewrite expressions using the properties of exponents, but there is no mention that this is connected to or building upon 8.EE.1.
- In Algebra I, Module 9, Task 1, the materials state that students will use “prior knowledge” to interpret data presented in a histogram and represent the same data with a box plot. Students previously displayed data in histograms and box plots with 6.SP.4, but this is not identified within the materials.
- In Geometry, Module 4, Task 3, the materials state, “The definition of similarity that students have been introduced to prior to this task is: Two figures are similar if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations”, and students extend this understanding to develop a new definition of similarity for polygons. The stated definition is established in 8.G.4, but is not identified within the materials.
- In Geometry, Module 4, Task 5, students apply the Pythagorean theorem to find missing side lengths and, conversely, determine whether given side lengths represent a right triangle. This builds upon 8.G.6 and 8.G.7, but these are not identified in the materials.
- In Algebra II, Module 3, Task 3, the materials build upon students’ understanding of division of whole numbers to support the development of polynomial long division without any identification that students would have developed that fluency in 6.NS.1.
- In Algebra II, Module 4, Task 5 indicates that students have compared and analyzed growth rates of functions but does not identify 8.F.2.
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.