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Focus & Coherence
Gateway 1 - Meets Expectations | 94% |
|---|---|
Criterion 1.1: Focus and Coherence | 17 / 18 |
The materials reviewed for enVisionMath A/G/A 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; allowing students to fully learn each standard; 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; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The materials partially meet expectations for attending to the full intent of the modeling process.
Criterion 1.1: Focus and Coherence
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 CCSSM).
The materials reviewed for enVisionMath A/G/A 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; allowing students to fully learn each standard; 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; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The materials partially meet expectations for attending to the full intent of the modeling process.
Indicator 1a
Materials focus on the high school standards.
Indicator 1a.i
Materials attend to the full intent of the mathematical content contained in the high school standards for all students.
The materials reviewed for enVisionMath A/G/A meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. All aspects of all-nonplus standards are addressed by the instructional materials of the series.
Examples of non-plus standards that are fully addressed in this series include:
N-CN.1: In Algebra 2, Topic 2, Lesson 2-4, students are introduced to imaginary numbers as a solution to a quadratic equation with no real roots and learn that complex numbers are identified in the form a+bi.
A-APR.3: In Algebra 1, Topic 9, Lesson 9-2, students are tasked with factoring polynomials to identify the zeros and then using those zeros to construct a graph of the polynomial. In Algebra 2, Topic 2, Lesson 2-3, students solve word problems by finding zeros of a quadratic function by factoring.
A-REI.10: In Algebra 1, Topic 2, Lesson 2-3, students use tables and equations to understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
A-REI.10: In Algebra 1, Topic 2, Lesson 2-3, students complete a table of values for a linear equation and graph the line through the points, demonstrating an understanding of the connection between the solutions to the equation and the graph. They extend that understanding by finding another solution using their graph and checking that point algebraically.
F-IF.6: In Algebra 1, Topic 5, Lesson 5-1, Topic 8, Lesson 8-1, and Topic 10, Lesson 10-1, students calculate and interpret the average rate of change for absolute value, quadratic, and square root functions over a specified interval. In Algebra 2, Topic 3, Lesson 3-1, students compare the average rate of change of a polynomial function over different intervals.
F-TF.8: In Algebra 2, Topic 7, Lesson 7-2, students use the Pythagorean Theorem with right triangles on the unit circle in various quadrants to derive the trigonometric functions and are asked to prove the Pythagorean identity.
G-CO.12: In Geometry, Topic 1, Lessons 1-2, students make formal geometric constructions. In Topic 2, Lessons 2-2, students construct a line parallel to a given line through a point not on the line, and in Topic 5, Lessons 5-1, they construct a perpendicular bisector.
G-SRT.7: In Geometry, Topic 8, Lesson 8-2, students explain and use the relationship between sine and cosine of complementary angles to find trigonometric ratios.
S-ID.4: In Algebra 1, Topic 11, Lesson 11-2, students recognize that there are data sets for which a procedure is not appropriate. In Algebra 2, Topic 11, Lesson 11-4, students use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
Indicator 1a.ii
Materials attend to the full intent of the modeling process when applied to the modeling standards.
The materials reviewed for enVisionMath A/G/A partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The materials omit the full intent of the modeling process for more than a few modeling standards across the courses of the series.
Each topic in the enVisionMath A/G/A 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 components.
Some of the modeling standards for which the full intent of the modeling process has been omitted include, but are not limited to:
N-Q.2: In Algebra 1, Topic 1, Lesson 1-3, Problem 45, students work on a performance task in which two individuals paint a wall. Although students determine when the painters have painted the same amount, they are not given the opportunity to create a conjecture and defend it as the rates are provided.
A-SSE.1a: In Algebra 1, Topic 7, Lesson 7-5, Problem 37, students use a quadratic expression to represent the area of a rectangular swimming space. Students do not formulate their own model as it is given and are told to factor the expression to determine possible dimensions of the swimming space.
A-SSE.3c: In Algebra 2, Topic 6, Lesson 6-2, Problem 25, students explain how a function could be rewritten to find a particular decay rate. Students do not have the opportunity to create their own models as they are told to rewrite an exponential growth function as part of their explanation.
G-MG.2: In Geometry, Topic 11, Lesson 11-2, Performance Task, Problem 27, students are given information on making candles and prompted to determine the smallest box that can be used to package a specific order of candles. Then, students are tasked with finding the volume of packing paper needed for the order. Students do not have opportunities to choose variables as the dimensions of each candle and box are presented.
F-IF.5: In Algebra 1, Topic 8, Lesson 8-1, students identify key features of a quadratic function. There are several application problems for students to practice determining the average rate of change over a specific interval. However, students do not interpret or evaluate the solutions. The questions do not provide multiple access points or various solutions.
F-IF.6: In Algebra 2, Topic 1, Lesson 1-1, students explain the meaning of the rate of change in the context of students jumping. Students do not have opportunities to choose variables, as the variables are identified for them.
S-ID.6b: In Algebra 1, Topic 3, Lesson 3-6, students graph residuals from a linear model of data, and in Lesson 8-4, students graph residuals from a quadratic model of data. Students do not complete the entire modeling process with this standard. In both lessons, students work with residuals in context, but students do not validate models or analyze results.
Examples where the materials intentionally develop the full intent of the modeling process across the series to address modeling standards include:
In Algebra 1, Topic 2, Mathematical Modeling in 3 Acts, students are presented with a situation in which height is measured in unconventional ways. Students watch a video that shows the height of a basketball player in terms of various objects. Students see a stack of cups being built next to him. As students attempt to figure out the basketball player’s height in foam cups, they have to formulate what information they would need. Data about smaller stacks of cups is given. Students utilize the information previously taught about linear functions. When students develop a plan, they complete computations. After they compute, students validate their findings when they view the final video which shows all the cups falling into place. Students report their findings compared to the final solution. (A-CED.1, A-CED.3 and A-CED.4)
In Geometry, Topic 11, Mathematical Modeling in 3 Acts, students explore and apply concepts related to surface area and volume. Students are presented with different packaging options for candles, and tasked with determining the packaging option with the least surface area for a constant volume. Students watch a video that shows 24 individually boxed candles. The smaller boxes are then packed inside one cardboard box. Students determine the dimensions of the package that has the least surface area. Students formulate a solution as they speculate how they could analyze the differences in surface area among the packages to find the one with the least surface area. Students compute a solution and think strategically to make sure they have found every possible set of dimensions for the packaging. They validate their results with each other to include ones not seen in the video. The final video shows the dimensions and surface area of each box. Students approach this solution using a variety of methods and report to each other. (G-GMD.3 and G-GMD.4)
In Algebra 2, Topic 6, Mathematical Modeling in 3 Acts, students explore and apply concepts related to exponential equations and functions. Students watch a video of an athlete performing a running drill. They extrapolate both the time and distance for a certain round of the drill. Students formulate how to determine how far the athlete runs in the twentieth round and how long it will take. Students determine what information they need and consider how their ideas might relate to exponential functions. Students interpret their findings, validate them with each other, and view the final video, which reveals the total time and distance. Students report their findings compared to the final solution. (F-LE.5 and S-ID.6a)
Indicator 1b
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
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 materials reviewed for enVisionMath A/G/A meet expectations for, when used as designed, allowing 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. Overall, the majority of the Algebra 1 materials address the WAPs, the Geometry materials provide a fairly even split between the WAPs and additional Geometry non-plus standards, and the Algebra 2 materials spend about 20% of the time extending the understanding of the WAPs and the rest further developing the other non-plus and plus standards.
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 1, Topic 1, students solve linear equations and inequalities. In Topic 9 and Algebra 2, Topic 2, students solve quadratic equations in one variable through graphing and completing the square (A-REI.B).
In Algebra 1, Topic 3, students determine whether a relation is a function by exploring domain and range (F-IF.A). In Algebra 2, Topic 1, students learn about the key features, and transformations of functions, while also applying functions to arithmetic sequences and series (F-IF.A and F-IF.B). In this topic, students also solve and create linear equations, inequalities, and systems (A-CED.A, A-REI.C, and A-REI.D). In Topic 2, students learn about the various forms of a quadratic function and explore procedures for finding solutions (A-CED.A, A-REI.B F-IF.B, and F-BF.B). In Topic 3, students learn about graphing and performing operations on polynomial functions (A-SSE.A, A-APR.A, F-IF.B, and F-IF.C). In Topic 4, students learn about how to graph rational functions, multiply, divide, add and subtract rational expressions, and solve rational equations (A-SSE.A, A-APR.D, and A-CED.A).
In Geometry, Topic 1, Lesson 1-7, students prove angle relationships within perpendicular lines using the transitive property of congruence. In Topic 2, Lesson 2-3, students use their knowledge of the linear pair postulate and exterior angle theorem to find the value of an angle that exists outside of a triangle. In Topic 6, Lesson 6-3, students find the lengths of two line segments using knowledge of vertical angles, alternate interior angles, and corresponding angles. (G-CO.C)
In Geometry, Topic 8, students use the right triangle trigonometry ratios to prove the Pythagorean Theorem. They use special right triangle relationships to solve right triangles from real-world scenarios. In Lesson 8-5, students apply right triangle trigonometry in scenarios of angle elevation and depression. (G-SRT.C)
In Algebra 2, Topic 11, Lesson 11-3, students find measures of center and spread, such as median, mean, interquartile range, and standard deviation, and compare data sets using statistical measures appropriate for the data's distribution (S-ID.A and S-IC.A).
Indicator 1b.ii
Materials, when used as designed, allow students to fully learn each standard.
The materials reviewed for enVisionMath A/G/A meet expectations for, when used as designed, allowing students to fully learn each non-plus standard. However, the instructional materials, when used as designed, do not enable students to learn a few of the non-plus standards.
The non-plus standards that would not be fully learned by students across the series include:
N-Q.3: In Algebra 1, Topic 1, Lesson 1-1, students decide which of three approximately equal numbers is most accurate and which is most appropriate given the context. In Topic 6, Lesson 6-4, students determine whether a set of data suggests linear or exponential growth. Students are not given sufficient opportunities to practice using a level of accuracy when reporting quantities throughout the series.
A-SSE.3c: In Algebra 1, Topic 6, Lesson 6-1, Example 5, students use the Product of Powers Property to solve equations with rational exponents. In Algebra 2, Topic 6, Lesson 6-2, Example 1, students rewrite an exponential function to identify a rate. Students are not given sufficient opportunities to demonstrate this understanding outside of examples.
F-LE.1a: In Algebra 1, Topic 6, Lesson 6-3, students compare how the linear function y = 3x grows over x with how the function y = 3x grows over the same values. The materials prove that linear functions grow by equal differences and that exponential functions grow by equal factors, but there is no opportunity for students to derive the proof on their own.
F-LE.3: In Algebra 1, Topic 8, Lesson 8-5, students compare linear, exponential, and quadratic graphs to determine which function will exceed the others in Example 3. Students are not given sufficient opportunities to demonstrate this understanding outside of examples.
G-GMD.1: In Geometry, Topic 11, Lesson 11-2, students use the properties of prisms and cylinders to calculate volume. However, students do not use dissection arguments and informal limits to fully learn this standard.
Indicator 1c
Materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The materials reviewed for enVisionMath A/G/A meet expectations for requiring students to engage in mathematics at a level of sophistication appropriate to high school. The 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 illustrate age-appropriate contexts for high school students include:
In Algebra 1, Topic 4, Lesson 4-3, students use systems of equations to determine the cost of a charter bus based on two field trip scenarios.
In Geometry, Topic 2, Lesson 2-2, students use properties of parallel lines and transversals to solve a problem involving a downhill skier maximizing their speed through a gate.
In Algebra 2, Topic 2, students write quadratics in various forms, identify key features, find zeros, and solve by all methods in contexts, including projectile motion in sports (soccer kicks, baseball hits, volleyball serves, golf swings, water balloons, catapults), jumps, dives, drone heights, and profit functions (e.g. tuition profit, sales profit).
Examples where the materials allow students to engage in the use of various types of real numbers include:
In Algebra 1, Topic 4, Lesson 4-2, students solve problems involving a lawn-mowing business and surfing lessons requiring students to manipulate and make sense of decimal answers.
In Geometry, Topic 8, Lesson 8-1, students use rational and irrational numbers when finding missing side lengths of right triangles.
In Algebra 2, Topic 3, Lesson 3-5, students find the zeros of polynomial functions, solutions include rational values, irrational values, and imaginary numbers.
Examples where the materials provide opportunities for students to apply key takeaways from Grades 6-8 include:
In Algebra 1, Topic 4, Lesson 4-5, students write and graph systems of linear inequalities as constraints for application-style questions (A-CED.3). These exercises build on solving and graphing solutions for inequality word problems with one variable (7.EE.4).
In Geometry, Topic 3, students apply their understanding of congruence and similarity through rotations, reflections, translations, and dilations (8.G.2) to learn about compositions of rigid motions and the effects on congruence (G-CO.6).
In Algebra 2, Topic 5, students solve and graph radical equations in one variable and demonstrate an understanding of extraneous solutions (A-REI.2 and F-IF.7). Students are building evaluating square roots of small perfect squares to represent solutions to equations (8.EE.2).
Indicator 1d
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 materials reviewed for enVisionMath A/G/A meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
Examples where the materials foster coherence within a single course include:
In Algebra 1, Topic 2, students write equations of linear functions given two points, a point, a slope, and a real-world description (A-CED.2). In Topic 4, students apply this skill to solving systems of linear equations, focusing on pairs of linear equations in two variables (A-REI.6).
In Geometry, Topic 6, Lesson 6-3, students use congruence and similarity criteria for triangles (G-SRT.5) to prove theorems about parallelograms (G-CO.11).
In Algebra 2, Topic 2, Lesson 2-5, the materials connect completing the square to find the maximum or minimum (A-SSE.3b) with graphing a quadratic function to show intercepts, maxima, and minima (F-IF.7a).
Examples where the materials foster coherence across courses include:
In Algebra 1, Topic 2, Lesson 2-4, students determine whether there is enough information to prove that two lines are parallel or perpendicular (G-GPE.5). In Geometry, Topic 6, Lessons 6-4 and 6-5 students use these skills to prove characteristics of quadrilaterals, including proving that a quadrilateral is a parallelogram (G-CO.11 and G-SRT.5).
In Algebra 1, Topic 11, Lessons 11-1, 11-2, and 11-3, students study data displays, center and variability, and histograms. In Geometry, Topic 12, students use histograms to display probability distributions and relative frequencies to conditional probability. In Algebra 2, Topic 11, students develop an understanding of statistical questions, hypothesis testing, random sampling methods, distribution of data sets, comparison of data values, and population parameters. In Topic 12, students develop an understanding of probability, conditional probability, and probability distributions (S-ID.A, S-ID.B, S.IC-A and S-IC.B).
In Algebra 1, Topic 9, students solve quadratic equations by completing the square, factoring, and using the quadratic formula. In Algebra 2, Topic 2, students use their knowledge of factoring to factor higher degree polynomials and solve equations that have complex solutions (N-CN.7, A-SSE.3a, A-SSE.3b, A-APR.3 and A-REI.4b).
Indicator 1e
Materials explicitly identify and build on knowledge from Grades 6-8 to the high school standards.
The materials reviewed for enVisionMath A/G/A meet expectations for explicitly identifying and building on knowledge from grades 6-8 to the High School Standards. The materials explicitly identify the standards from grades 6-8 in the Math Background Coherence section for each topic in the Teacher’s Edition. This information appears routinely in the design of the teacher materials but not in the student and family materials.
Examples where the teacher materials explicitly identify content from Grades 6-8 and build on them include:
In Algebra 1, Topic 1, Lesson 1-6, students build on their knowledge of solving inequalities in the form px+q>r or px+q
In Algebra 1, Topic 1, students build on their knowledge of one variable inequalities (7.EE.4b) in Lessons 1-5 and 1-6 by solving and graphing compound and absolute value inequalities (A-REI.3).
In Algebra 1, Topic 3, students extend their exploration of linear, nonlinear, and the key features of linear functions (8.F.1, 8.F.2, and 8.F.4) by determining the domain and range of a linear function, writing linear function rules, and transforming linear functions. (F-IF.1, F-IF.2, F-IF.5, and F.BF.3).
In Geometry, Topic 12, students examine probabilities of multiple-outcome events (S-CP.6 and S-CP.7) to expand on an understanding of basic theoretical and experimental probability (7.SP.6 and 7.SP.7).
In Algebra 2, Topic 11, students use box plots and histograms to evaluate data distributions and determine the shape of the graph, find the standard deviation, and interpret data to determine if the data is skewed and/or has outliers (S-ID.A) to build on their knowledge of using dot plots, box plots, and histograms to represent data and find measures of central tendency (6.SP.2 and 6.SP.4).n
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.