High School - Gateway 1
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Focus & Coherence
Gateway 1 - Meets Expectations | 77% |
|---|---|
Criterion 1.1: Focus & Coherence | 14 / 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 enVision Integrated Mathematics meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content for all students; spending the majority of time on content widely applicable as prerequisites; engaging students in mathematics at a level of sophistication appropriate to high school; and making meaningful connections in a single course and throughout the series. The materials partially meet expectations for the remaining indicators in Gateway 1, which include: attending to the full intent of the modeling process; allowing students to fully learn each standard; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.
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 enVision Integrated Mathematics meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials address most aspects of all non-plus standards across the courses of the series.
Examples of non-plus standards fully addressed by the series include, but are not limited to:
- N-RN.1,2: In Mathematics I, Lesson 5-1, multiple examples address square root and rational relationships and identify the product of powers property. Students work with the properties throughout the series, as in Mathematics II, Lesson 1-2, and Mathematics III, Lesson 4-1.
- A-SSE.3c: In Mathematics I, Lesson 5-3, students work with changing the format as indicated by the standard, and in Mathematics II, Lesson 1-3, students work with basic interest problems. In Mathematics III, Lesson 5-2, Example 1, students apply the power rule to show the change of form of the equations, and Example 2 includes rewriting an expression with exponents using the properties to transform the expressions.
- F-IF.4: In Mathematics I, Lesson 5-2, Example 1 defines asymptotes, the end behavior of a graph as x approaches negative infinity, and the behavior of a graph as y approaches 0. Mathematics I, Lesson 5-5, Example 3 discusses end behavior as the graph approaches -1. In Mathematics II, Lesson 3-3, multiple examples address y-intercepts, vertex, axis of symmetry, and a context problem. Mathematics II, Lesson 6-1, Example 3 addresses increasing absolute value functions and the axis of symmetry, and Mathematics III, Lesson 6-4, Example 1 addresses the graph of a periodic function.
- G-CO.7: Mathematics I, Lesson 9-3, Example 1 includes a video that explains how students prove triangles are congruent through rigid motions, and Theorem 9-4 makes this explicit.
Non-plus standards partially addressed by the series include:
- F-LE.1a: In Mathematics I, Lessons 3-2 (Linear) and 5-2 (Exponential), students are shown linear and exponential functions growing by equal differences and equal factors, but no proof is included in the materials for either type of function.
- G-CO.2: In Mathematics II, Lesson 9-1, students work with dilations, but no evidence was found where students are shown or compare the transformations that preserve distance and angle measures to those that do not.
- G-CO.8: In Mathematics I, Lessons 9-3 and 9-4, rigid motions are referenced, but there was no evidence that the students had to explain the connection between the criteria for triangle congruence and the definition of congruence in terms of rigid motions.
- G-C.3: In Mathematics III Lesson 9-1, Part A of the lesson, (marked "Extension" and may not be required) students construct the circles but do not prove properties for quadrilaterals. They are not asked for or shown proof in Part B or Part C of the lesson.
- G-GPE.7: In Mathematics I, Lesson 9-7, students use coordinates to compute perimeter and area of polygons, however, there was no evidence found of computing perimeter or area using coordinates of a rectangle.
- G-GMD.1: In Mathematics II, Lessons 13-2 and 13-3, students work with the formulas for the volume of a cylinder, pyramid, and cone and apply Cavaleri's Principle to pyramids and cones. There was no evidence in the series where circumference and area of a circle were addressed.
- S-ID.4: In Mathematics III, Lesson 8-3, multiple examples introduce students to the appropriateness of procedures for estimating population proportions. The series does not address using technology and calculators to estimate the areas under the Normal Curve.
Standard not addressed by the series include:
- A-REI.10: No evidence was found in the series with regard to a curve (or a line) representing all of the solutions plotted in the coordinate plane to an equation in two variables.
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 enVision Integrated Mathematics partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials omit the full intent of the modeling process for more than a few modeling standards across the courses of the series.
Each topic of the series contains “Mathematical Modeling in 3 Acts” and STEM projects. In each lesson, students are posed a problem, usually by watching a video. Students develop questions of their own, formulate a conjecture, and explain how they arrived at the conjecture. In most of the tasks, the needed information is not given, and students determine what information is essential. Students compute a solution for the problem and interpret their results. Students are guided through validating their conjecture and considering reasons why their answers might differ. Students engage in the full modeling process within the “Mathematical Modeling in 3 Acts” and STEM projects. However, several modeling standards are not addressed within these 3 Acts and STEM projects.
Some of the modeling standards for which the full intent of the modeling process has been omitted include:
- N-Q.1 and A.SSE.3c: In Mathematics I, Lesson 1-3, students use a formula to find the dimensions of the flag with the variable given. Students do not make conjectures and are not asked to validate or interpret answers. No evidence was found where students have the opportunity to engage in the full modeling cycle with these standards.
- N-Q.2: In Mathematics I, Lesson 1-2, students have opportunities to define quantities, formulate equations, and compute and interpret results. Students answer specific questions that guide them but do not conjecture or validate their choices.
- A-SSE.1b: In Mathematics I, Lesson 5-3, Practice and Problem Solving, Problem 27, students use an exponential growth function to model and explain how they found the solution, but the students do not make a conjecture because the conjecture is already given in the problem. In Mathematics II, Topic 1, students use tools to find an exponential model for the given data but do not interpret results.
- A-SSE.3b: In Mathematics II, Lesson 5-2, Performance Task, students formulate a function that represents increasing area by adding to given dimensions. Students compute using the function to find the new dimensions. In Mathematics II, Lesson 1-3, Practice & Problem Solving, Problems 14 and 30, are described as "model with mathematics." However, students formulate and compute, but students do not make assumptions, predictions, or interpret results.
- A-SSE.4: In Mathematics III, Lesson 5, students formulate and compute in many problems aligned to this standard. Students do not encounter extraneous solutions and do not summarize conclusions, make assumptions and conjectures, or validate answers.
- A-REI.11: In Mathematics I, Lesson 4-1, students are led through the formulation of an equation set and creation of a graph which they interpret. In the Try it! problems, students repeat the steps themselves. In Mathematics III, Lesson 1-5, Practice and Problem Solving, Problem 32, students solve a system of equations related to revenue and expenses. However, students are provided what to do rather than engaging in the complete modeling cycle in either course.
- F-IF.5: In Mathematics I, Lesson 3-2, Practice and Problem Solving, Problem 29, students are directed what to do in Part A to solve the main question written in Part B. Students do not make conjectures, but are directed in how to proceed.
- F-IF.7a: In Mathematics II, Lesson 3-3, Performance Task, Problem 42, students use the information given to write a quadratic model. Students are provided with a quadratic function, a quadratic graph, and data points for a third quadratic relationship. Students identify maximum profits for each scenario and explain their results. Students formulate a function for the third model, compute and interpret maximized results for all three situations, and explain/validate their results. However, each situation has one solution, and students do not predict the best model or justify their responses.
- G-GPE.7: In Mathematics I, Lesson 9-7, Practice and Problem Solving, Problem 26, students find one solution. In Practice and Problem Solving, Problem 28, students formulate, compute, interpret, validate and report, but since there is one specific answer, students do not revise their work.
Examples where the materials intentionally develop the full intent of the modeling process across the series to address modeling standards include:
- In Mathematics I, Topic 2, Mathematical Modeling in 3 Acts, students determine a basketball player's height in unconventional ways, which culminates in determining the height based on the height of foam cups. Students use their knowledge of linear functions and data given about smaller stacks of cups to determine his height. Students make predictions, report their findings, and compare their solutions to the findings presented in the final video. Students represent the relationship between the number of cups and the height of the stacks to generate more data. (A-CED.1,3,4)
- In Mathematics III, Topic 8, STEM project, students conjecture, make a model, calculate, justify, validate, and report their conclusions. Students use statistical surveys to decide how best to use public spaces. Students plan a new public space by deciding how to gather data, analyze the data, and report the conclusions. (S-IC.B)
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 enVision Integrated Mathematics 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.
- In Mathematics I, the materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. The majority of the lessons in Mathematics I address the WAPs, and there were only a few lessons that did not include a WAP.
- In Mathematics II, the materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, and Geometry. The majority of the lessons in Mathematics II address the WAPs, and there are fewer lessons that focus on WAPs compared to Mathematics I.
- In Mathematics III, the materials do not spend the majority of the lessons on the WAPs. The materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, and Statistics and Probability.
Examples of how the materials allow students to spend the majority of their time on the WAPs include:
- Throughout the series, students get several opportunities to work with standards from F-IF. In Mathematics I, Topic 1, students solve linear equations and inequalities in one variable and work with absolute value equations. In Topic 2, students graph, solve, and manipulate linear equations. In Mathematics II, Topic 4, students solve quadratic equations by factoring, using square roots, graphs, and tables. In Mathematics III, Topic 5, students convert equations between logarithmic and exponential forms, solve logarithmic and exponential equations, graph logarithmic equations, and explore properties of logarithms.
- In Mathematics I and II, students engage with G-SRT.5 by exploring the properties of similarity. In Mathematics I, Lessons 9-2 through 9-6, students explore the relationship between similarity and triangle congruence. In Mathematics II, Lessons 7-5, 8-1, 8-3 through 8-6, and 9-2 through 9-4, students explore similarity relationships with quadrilaterals.
- The series addresses A-SSE.2 throughout Mathematics II and III in a variety of places with a variety of practice and expressions (exponential, cubic, quadratic, etc.). In Mathematics II, Lesson 1-3, Example 2 includes using the structure to rewrite an interest formula. In Lesson 2-7, Example 1 explores perfect-square trinomials and writing equivalent expressions for those containing perfect-square trinomials. In Lesson 5-5, Example 2 includes the use of polynomial identities to multiply polynomials after rewriting the expression using its structure. In Mathematics III, Lesson 5-2, students use the structure of exponential functions to write and rewrite models.
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for (when used as designed) letting students fully learn each non-plus standard. The instructional materials for the series (when used as designed) do not enable students to fully learn some of the non-plus standards. For the most part, students do not have the opportunity to independently prove theorems throughout the series.
The non-plus standards that would not be fully learned by students across the series include:
- N-Q.1: Mathematics I, Lessons 1-3 and 9-7, and Mathematics II, Lesson 6-1 are examples of lessons throughout the series in which units and scales are provided for students, but students do not “choose and interpret units consistently in formulas.”
- N-Q.3: In Mathematics III, Lesson 9-1, students perform basic constructions using a ruler in Example 1, but students do not connect the constructions with taking appropriate measurements. Throughout the series, a level of accuracy appropriate to limitations on measurement is not discussed or modeled, and students are not given sufficient opportunities to practice choosing a level of accuracy when reporting quantities.
- A-APR.4: In Mathematics II, Lesson 5-5, students prove the sum of cubes and difference of cubes' identities. Two other identities are proven, but students do not use them to describe numerical relationships.
- A-REI.1: In Mathematics I, Lesson 1-2, students explain each step of solving a simple equation in an example, but there are no other opportunities for students to practice constructing a viable argument with other simple equations.
- A-REI.5: In Mathematics I, Lesson 5-4, students practice solving systems of equations by elimination; however, students do not prove (given a system of two equations in two variables) that replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- F-IF.3: In Mathematics I, Lesson 3-4, Example 1 shows a sequence being a function. However, there are no other opportunities found for students to recognize that sequences are functions. In Mathematics III, Lesson 1-4, students determine whether each sequence is arithmetic, but students do not develop a connection between sequences and functions.
- F-IF.7: Throughout the series, students do not use technology to create graphs, as graphs and screenshots from technology are consistently provided for students.
- F-LE.3: In Mathematics II, Lesson 3-5, students compare linear and exponential graphs to determine which function will exceed the other in one problem, and in Mathematics III, Lesson 5-3, Problem 17, students determine when an exponential function will exceed a linear function and a quadratic function.
- F-TF.8: In Mathematics II, Lesson 9-7, students use the Pythagorean identity to find trigonometric values, and in Mathematics III, Lesson 6-1, Example 2, students use one trigonometric ratio to find another trigonometric ratio using the Pythagorean Theorem. In Mathematics III, Lesson 6-3, students also use the Pythagrean identity, but students do not prove the Pythagorean identity.
- G-SRT.2: In Mathematics II, Lessons 9-1and 9-2, students determine whether or not triangles are similar. Students do not use similarity transformations to explain the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Indicator 1c
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed for enVision Integrated Mathematics 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.
Some examples where the materials illustrate age-appropriate contexts for high school students include:
- In Mathematics I, Lesson 4-2, students encounter problems about a lawn mowing business and going to an amusement park.
- In Mathematics II, Topic 4, STEM Project, students use quadratic functions to model parabolic vertical motion and design an appropriate t-shirt launcher.
- In Mathematics III, Topic 4, Problem 34, students write a function for the cost of hiring a DJ for an event.
Some examples where students apply key takeaways from Grades 6-8 include:
- In Mathematics I, Lesson 2-1, Model and Discuss, students use proportional relationships to determine which payment plan would be better.
- In Mathematics I, Lesson 3-2, Problem 30, students write a linear function to model the data and determine the total cost of a job that took 4 hours and 15 minutes of labor.
- In Mathematics II, Lesson 1-1, Problems 8 and 9, students order a set of numbers from least to greatest including integers, rational numbers, and irrational numbers.
- In Mathematics II, Lesson 8-2, Problem 22, students use a diagram of a map to determine a 2 mile run for the team, applying a conversion of yards to miles and interpreting the map.
- In Mathematics II, Lesson 9-1, Explore and reason, students use similarity understanding to build dilated figures.
- In Mathematics II, Lesson 10-2, students calculate probabilities to develop understanding of the probability of mutually exclusive events.
- In Mathematics III, Lesson 8-7, students determine if games of probability and chance are fair and determine percentages based on probabilities.
Some examples where the instructional materials use various types of real numbers include:
- In Mathematics I, Lesson 1-1, students solve linear equations including all types of rational numbers (positive and negative integers, decimals, and fractions) within examples and practice problems.
- In Mathematics I, Lesson 2-2, the problem set has integers and fractions in the equations for students to graph.
- In Mathematics I, Lesson 6-3, students calculate midpoints for number sets that include negative numbers, decimals, and fractions.
- In Mathematics II, Lesson 4-4, students solve quadratic equations including irrational numbers, and Lesson 5-1 includes operations with complex numbers.
- In Mathematics III, Lesson 2-6, students discuss the rational root theorem for polynomial functions, along with irrational and complex roots.
- In Mathematics III, Lesson 4-3, students solve equations containing square and cube roots.
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 enVision Integrated Mathematics meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials make meaningful connections within and across courses, and the materials provide guidance so that multiple lessons are not repeated across the courses of the series.
Examples of the materials making meaningful connections within courses include:
- In Mathematics I, Lesson 1-1, students use linear equations in one variable to solve problems (A-CED.1). In Lessons 2-1 and 2-2, students use linear equations in two variables to represent relationships between two quantities and graph those relationships on the coordinate plane (A-CED.2). In Lesson 4-4, students represent inequalities in two variables and solve problems in context (A-CED.3).
- In Mathematics II, Lessons 7-1, 7-2, and 7-3, students prove relationships about lines and angles (G-CO.9), and in Lessons 8-3 and 8-4, students use those angle relationships to prove properties of parallelograms(G-CO.11). In Lesson 9-3, students prove triangle similarity relationships using angle and line relationships (G-CO.10).
- In Mathematics III, Lesson 1-1, students review the key features of functions (F-IF.7). In Lesson 1-3, students graph absolute value and piecewise functions by identifying the maximum value, minimum value, and other key features of the graph (F-IF.7b). In Lesson 2-5, students find the zeros of polynomials and describe their behavior at each zero (F-IF.7c).
Examples of the materials making meaningful connections across courses include:
- In Mathematics I, Lesson 5-2, students graph and label key features of exponential functions. In Mathematics II, Lesson 3-1, students identify key features of quadratic functions, and in Mathematics III, Lesson 2-1, students determine the end behavior of polynomial functions (F-IF.4).
- In Mathematics I, Lesson 3-3, students describe how adding k affects the graph of a line. In Mathematics II, Lesson 3-1, students describe how the leading coefficient of a quadratic equation affects the graph of that equation. In Mathematics III, Lesson 5-1, students identify the effect on graphs by replacing values for x (F-BF.3).
- In Mathematics II, Lesson 6-1, students use their knowledge of linear distance to develop an understanding of absolute value as a function and that each point has a corresponding point equidistant from the vertex of the graph of an absolute value function. In Mathematics III, Lesson 6-1, students use similar triangles, ratio reasoning, and trigonometric identities to develop trigonometric functions (F-IF.7b; F-TF.2).
- In Mathematics I, Chapter 9, students prove congruent relationships in triangles and other geometric figures. In Mathematics II, Lesson 9-2, students use similarity transformations to determine if polygons and triangles are similar. In Mathematics III, lesson 6-1, students apply their knowledge of triangle similarity and congruence to side ratios in right triangles (G-SRT.5,6).
- In Mathematics I, Lesson 1-4, students write linear equations for different contexts. In Mathematics II, Lesson 3-4, students create a quadratic equation with two or more variables given a context. In Mathematics III, Lesson 1-5, students create linear, absolute value, and quadratic equations from given contexts (A-CED.1,2).
Below is the list of lessons in Mathematics III that are duplicated from previous courses in the series. In the Teacher Resources for each course, there is a document entitled enVision Integrated CC Pathway. The document includes the lessons that should be included in each course and the courses from which the repeated lessons should be omitted. The guidance provided by the document eliminates disruptions to the coherence of the materials that would occur if lessons were repeated from one course to another.
- Mathematics III Lesson 9-1 is a duplicate of Mathematics I Lesson 6-2 (N-Q).
- Mathematics III Lesson 5-2 is a duplicate of Mathematics II Lesson 1-3 (A-SSE.3).
- Mathematics III Lesson 2-3 is a duplicate of Mathematics II Lesson 5-5 (A-APR.4, A-REI.4).
- Mathematics III Lesson 9-4 is a duplicate of Mathematics II Lesson 11-2 and Lesson 10-1 is a duplicate of Mathematics II, Lesson 12-1. (G-CO.1)
- Mathematics III Lesson 9-3 is a duplicate of Mathematics II Lesson 11-1. (G-CO.10)
- Mathematics III Lessons 10-1 through 10-5 are duplicates of Mathematics II Lessons 12-1 through 12-5. (G-C.2,4,5)
- Mathematics III Lessons 9-5 and 9-6 are duplicates of Mathematics II Lessons 11-3 and 11-4. (G-GPE.1-3)
- Mathematics III Lessons 11-1 through 11-4 are duplicates of Mathematics II Lessons 13-1 through 13-4. (G-GMD.1-3)
- Mathematics III Lessons 11-2 and 11-4 are duplicates of Mathematics II Lessons 13-2 and 13-4. (G-MG.1,2)
Statistics and Probability standards (S-ID.C, S-IC, S-CP, and S-MD) are primarily addressed in Mathematics I or Mathematics III and are not often connected to other categories of standards within or across courses.
Indicator 1e
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials do not explicitly identify content from Grades 6-8. There are limited places where “6-8” is written, but specific standards are not mentioned in the teacher or student materials.
Each chapter throughout the series has Math Background Coherence, which includes Looking Back, This Topic, and Looking Ahead. In these sections, concepts and skills from Grades 6-8 are mentioned, but domains, clusters, or standards are not explicitly identified.
Some examples where the materials do not explicitly identify content from Grades 6-8 include:
- In Mathematics I, Teacher Edition, the Topic 6 Overview states, “In Grade 8, students used lines, segments, rays, and angles.”
- In Mathematics II, Teacher Edition, the Topic 3 Overview indicates the grade 8 content of analyzing graphs of functions as background knowledge for connecting to quadratic functions.
- In Mathematics III, Teacher Edition, the Topic 8 Overview states, “In Grade 6 students used dot plots, box plots, and histograms to represent data.”
Some examples where the materials make connections between Grades 6-8 and high school concepts and build on students’ previous knowledge include:
- In Mathematics I, Topic 3 Overview, Teacher Edition, students explore linear and nonlinear functions as well as key features of linear functions that include slope and rate of change. Students will extend their understanding to determine the domain and range of linear functions, write linear function rules, and transform linear functions.
- In Mathematics I, Topic 10 Overview, Teacher Edition, students use dot plots, boxplots, and histograms to represent data. Students will extend their understanding of these data displays to interpret and compare data.
- In Mathematics II, Topic 2 Overview, Teacher Edition, students factored expressions by identifying the greatest common factor and using the distributive property (in Grade 6). This will be used in Topic 2 when factoring polynomials, including trinomials with binomial factors.
- In Mathematics II, Topic 7 Overview, Teacher Edition, students studied the radius of the circle (in Grade 7), and in grade 8, they learned to identify congruent figures using transformations. In Topic 7, students study the relationships and side lengths in a single triangle. In Mathematics III, students will extend the relationship of triangles and circles when they make sense of trigonometric ratios.
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.