## Alignment: Overall Summary

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for Alignment to the CCSSM. The materials meet expectations for Focus and Coherence as they show strengths in: 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 the expectations for Rigor and Mathematical Practices as they partially meet the expectations for Rigor and Balance and partially meet the expectations for Practice-Content Connections. Within Rigor and Balance, the materials did show strengths with providing students opportunities for developing procedural skills and balancing the three aspects of Rigor, and within Practice-Content Connections, the materials showed strength in developing seeing structure and generalizing (MPs 7 and 8).

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## Gateway 1:

### Focus & Coherence

0
9
14
18
14
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

## Gateway 2:

### Rigor & Mathematical Practices

0
9
14
16
10
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

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## Gateway 3:

### Usability

0
21
30
36
N/A
30-36
Meets Expectations
22-29
Partially Meets Expectations
0-21
Does Not Meet Expectations

## The Report

- Collapsed Version + Full Length Version

## Focus & Coherence

### Criterion 1a - 1f

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).
14/18
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Criterion Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series 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 the 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.*
0/0

### Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
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Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics 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 address the full intent of many of the non-plus standards, and there are a few instances where aspects of the non-plus standards are not addressed across the courses of the series.

The following are examples for which the materials attend to the full intent of the standard:

• N-RN.2: In Algebra II, Unit 1, Lesson 4, students find the product of monomials by multiplying numerical coefficients and adding exponents. In Algebra II, Unit 8, Lesson 3, students rewrite expressions in simplest form by finding the quotient and the product of expressions involving rational exponents. In Algebra II, Unit 8, Lesson 5, students rewrite expressions with rational exponents and radicals.
• A-APR.4: In Algebra II, Unit 10, Lesson 4, students prove the following polynomial identities: $$x^2-y^2=(x - y)(x + y)$$, $$(a+b)^2=a^2+ 2ab +b^2$$, $$(x+c)(x+d) = x^2+(c+d)x+cd$$, $$x^3-y^3=(x-y)(x^2+xy+y^2)$$, and $$(x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2$$. After proving the identities, students evaluate products and generate Pythagorean triples.
• A-REI.6: In Algebra I, Unit 5, Lesson 1, students graph a system of linear equations and use a table on a graphing calculator to find solutions. In Algebra I, Unit 5, Lesson 2, students solve systems of equations by substitution, and in Lesson 4, students solve systems of linear equations by elimination.
• F-IF.4: In Algebra I, Unit 3, Lesson 4, students interpret the minimum and maximum values of a function and sketch a graph showing the minimum and maximum values. In Algebra I, Unit 3, Lesson 5, students identify key features of a graph including relative maximum and/or minimum values, x- and y- intercepts, and intervals where the function is increasing and/or decreasing. In Algebra II, Unit 10, Lesson 1, students sketch a graph using a verbal description of end behavior.
• F-BF.3: In Algebra I, Unit 8, Lesson 3, students identify the effects on a graph by comparing $$f(x) =x^2$$ and $$g(x) =(x-2)^2 -4$$ using a graphing calculator. In Algebra I, Unit 8, Lesson 5, students identify the effect on a parabola by replacing f(x) with f(kx). In Algebra I, Unit 9, Lesson 3, students identify the transformations of square root functions by replacing $$y =\sqrt x$$ with $$y =\sqrt {x+4}+2$$. Students also identify transformations of absolute value functions by replacing y = |x| with y = |x + 3| - 2. In Algebra II, Unit 7, Lesson 3, students identify the transformation of a graph by replacing f(x) with kf(x). In Algebra II, Unit 10, Lesson 5, students recognize even and odd functions from graphs.
• G-CO.12: In Geometry, Unit 4, Lesson 2, students create a copy of angles and construct parallel lines using a straightedge and a compass. In Geometry, Unit 4, Lesson 3, students construct the midpoint of a line segment and a perpendicular bisector of a segment. In Geometry, Unit 4, Lesson 4, students construct an angle bisector.
• G-GPE.4: In Geometry, Unit 8, Lesson 6, students use coordinates to determine if a triangle is a right triangle. In Geometry, Unit 5, Lesson 7, students use coordinates to prove that a triangle is an isosceles triangle, and students explain if a point lies on a circle given the center and radius of the circle.
• S-ID.6a: In Algebra I, Unit 10, Lesson 6, students construct a scatter plot and determine the line of best fit to answer questions about the data. In Algebra I, Unit 10, Lesson 7, students use a calculator to generate the line of best fit and use data to answer questions. In Algebra I, Unit 10, Lesson 8, students determine if the line of best fit for a set of data is a linear, exponential, or quadratic function.

The materials attend to some aspects, but not all, of the following standards:

• A-REI.4a: In Algebra I, Unit 8, Lesson 4, students complete the square to transform quadratic equations into vertex form. The materials do not contain the derivation of the quadratic formula. In Algebra I, Unit 9, Lesson 6, the materials state that a proof or derivation of the quadratic formula "is beyond the scope of this course."
• F-IF.8a: In Algebra I, Unit 8, Lesson 4, students complete the square in order to determine the extreme value. In Algebra I, Unit 8, Lesson 6, students factor quadratics to show zeros. The materials do not provide opportunities to show symmetry of the graph and interpret in terms of a context.
• F-LE.3: In Algebra I, Unit 6, Lesson 8, students use tables and graphs to observe how a quantity increasing exponentially exceeds a quantity increasing linearly. The materials do not provide opportunities to observe how a quantity increasing exponentially exceeds a quantity increasing quadratically or as a polynomial function.
• S-ID.4: In Algebra II, Unit 13, Lesson 3, students calculate standard deviation for a population that is normally distributed. Students calculate the population percentage of math scores when given the mean score and the standard deviation. In Algebra II, Unit 13, Lesson 4, the materials state that calculators and tables can be used to calculate probabilities under the normal curve. The materials also introduce z-scores and have students calculate z-scores. The materials do not address that there are data sets that do not fit a normal distribution and z-scores could not be used to estimate population percentages.

The following standard was not addressed across the courses of the series:

• S-IC.6: The materials do not include data reports for students to evaluate.

### Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.
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Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials include various aspects of the modeling process in isolation or combinations, but opportunities to engage in the full modeling process are absent from the materials.

Examples of the materials applying aspects of the modeling process, but not the full modeling process, to modeling standards include, but are not limited to:

• A-SSE.1a: In Algebra I, Unit 8, Lesson 2, students calculate the cost per computer produced at a factory if 50 computers were produced in a day and interpret their answer in terms of the scenario. Students determine if the cost has a minimum or maximum value and use a calculator to validate their explanation. Students also determine if the function has any real zeroes. Throughout the scenario, students explain their thought process when finding solutions. Students do not formulate an equation to represent the situation, as an equation is given within the problem.
• A-CED.3: In Algebra I, Unit 5, Lesson 8, students encounter a scenario where a person works as a carpenter and a website designer. The scenario states, “He can work at most 50 hours per week and makes $35 per hour as a carpenter and$75 an hour as a website designer.” Students formulate a system of inequalities to model the number of hours needed to earn $2350 per week and work at least 10 hours as a carpenter. Students do not define the variables to represent the number of hours worked as a carpenter and the number of hours worked as a website designer. Students calculate the maximum amount of money made in a week using the system of inequalities and validate the coordinates of the intersection of the graphs. Students interpret what the value of the hours worked as a carpenter means based on the solution of the system of inequalities, but students do not report their reasoning for summarizing their conclusions. • F-LE.1c: In Algebra I, Unit 6, Lesson 3, students encounter a scenario of a person, Helmut, who is walking to a windmill. The problem states, “On his first trip he walks half the distance to the windmill. On his next trip he walks half of that is left. On each consecutive trip, he walks half of the distance he has left.” Students are instructed to model the distance Helmut has remaining. Students formulate an equation to represent the distance after n-trips and calculate the distance of Helmut after 6 trips. Students verify their solution by providing calculations as justification. Students interpret why Helmut will not reach the windmill, but students do not report their reasoning for summarizing their conclusions. • G-SRT.8: In Geometry, Unit 5, Lesson 6, students encounter a 20-foot ladder leaning against a building reaching a window 8 feet off the ground. Students also are given the constraint that if the base of the ladder is more than 15 feet from the bottom of the building, the ladder will be unstable. Students determine if the ladder will be unstable and provide their explanation, but students do not validate their answers. • S-CP.6: In Algebra II, Unit 12, Lesson 4, students use a survey of how commuters travel to work to answer questions. Students calculate the probability of a randomly selected person taking the train to work in Los Angeles and a randomly selected person living in New York City taking a car to work. Students interpret if it is, “more likely that a person who takes a train to work lives in Chicago or more likely that a person who lives in Chicago will take a train to work.” Students report their reasoning and support their work with conditional probabilities. Students do not formulate or validate solutions within the problem. The materials did not apply aspects of the modeling process to modeling standards F-IF.7c and G-GPE.7. ### 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. 0/0 ### 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. 2/2 + - Indicator Rating Details The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics 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. The Algebra I and Algebra II materials address the WAPs for the majority of the courses, and the Geometry materials spend less than the majority of time addressing the WAPs. Examples of students engaging in WAPs include: • In Algebra I, Unit 8, Lesson 4, students write equations in vertex form by completing the square. Students complete the square to identify the coordinate of the maximum or minimum. In Algebra I, Unit 8, Lesson 5, students complete the square to identify the coordinate of the maximum or minimum and identify whether the calculated coordinate is the maximum or minimum (A-SSE.3b). • In Algebra I, Unit 11, Lesson 4, students determine if a linear equation or an exponential equation is needed for a scenario. Students write a linear equation and an exponential equation for the scenario. Students then determine which model would better represent the situation and justify their decision. In Algebra I, Unit 4, Lesson 6, students calculate the average rate of change for a deer population for several intervals. Students describe why the average rate of change represents a linear function (F-LE.1b). • In Geometry, Unit 7, Lesson 8, students algebraically prove the Side Splitter Theorem and explore the converse of the Side Splitter Theorem to verify it as true. In Geometry, Unit 7, Lesson 12, students prove the Pythagorean Theorem using similarity (G-SRT.4). In Geometry, Unit 3, Lesson 3, students use congruence criteria to prove triangles are congruent. In Geometry, Unit 7, Lessons 4, 5, and 6, students use similarity criteria to solve problems, and in Lesson 4, students use transformations to show triangles are similar. In Lesson 5, students use similarity criteria to prove triangles are similar, and in Lesson 6, students use similarity criteria to prove triangles are similar, as well as, solve for missing lengths and angles (G-SRT.5). • In Algebra II, Unit 13, Lesson 2, students determine population parameters of scenarios including mean, standard deviation, and interquartile range, and in Lesson 5, students use the sample mean in order to estimate the population mean when given a random sample from a population (S-IC.1). ### Indicator 1b.ii The materials, when used as designed, allow students to fully learn each standard. 2/4 + - Indicator Rating Details The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for, when used as designed, letting students fully learn each non-plus standard. The non-plus standards that would not be fully learned by students across the series include: • N-RN.3: In Algebra I, Unit 9, Lesson 2, students investigate the sums and products of irrational and rational numbers. Students determine the product of a non-zero rational number and an irrational number is an irrational number. Students determine the sum of a rational number and an irrational number is an irrational number. Students do not explain why the sum or product is rational or irrational. • N-Q.1: In Algebra I, Unit 10, Lesson 1, students construct a histogram to represent a random survey of 100 cars and the fuel efficiency in miles per gallon. Students choose the scale and intervals of the graph and also label the axes appropriately. In Algebra I, Unit 6, Lesson 9, students graph the first six terms of a sequence on a coordinate plane. Students are given the scale and identification of the x-axis, and students scale and identify the y-axis. In Algebra II, Unit 7, Lesson 4, students sketch a graph of an absolute value function. Students are given the scales of the x-axis and y-axis. Students do not have further opportunities to choose the scale(s) for graphs and data displays. • A-CED.1: In Algebra I, Unit 2, Lesson 5, students create and solve equations from verbal expressions. In Algebra I, Unit 2, Lesson 9, students create and solve linear inequalities to solve problems. In Algebra I, Unit 8, Lesson 8, students create quadratic equations representing scenarios and then solve. In Algebra II, Unit 4, Lesson 6, students create and solve exponential equations to represent growth and decay. Students do not create simple rational equations to solve problems. • A-REI.11: In Algebra I, Unit 7, Lesson 6, students solve absolute value, linear, and quadratic equations using graphs, primarily on a graphing calculator, to understand how the x-coordinates of the points where the graphs intersect are the solutions of the equation f(x) = g(x). Students do not encounter cases where f(x) and/or g(x) are logarithmic equations. • F-IF.9: In Algebra I, Unit 8, Lesson 4, students compare two quadratic functions to determine the lower minimum value. In Algebra II, Unit 7, Lesson 4, students compare the graphs of two functions. Students do not have other opportunities to compare functions represented in different ways. • F-BF.1b: In Algebra II, Unit 4, Lesson 14, students build complex functions using exponential functions. In Algebra II, Unit 3, Lesson 4, students create an equation representing profit by subtracting revenue from total cost, and students combine two linear equations algebraically. Students do not have further opportunities to combine standard function types using arithmetic operations. • F-BF.2: In Algebra II, Unit 5, Lesson 2, students write arithmetic and geometric sequences both recursively and explicitly. Within the same lesson, students translate from recursive to explicit form. Students do not translate from explicit to recursive. • F-TF.5: In Algebra II, Unit 9, Lesson 9, students use a trigonometric model to represent tides in a bay. Students determine values to substitute into the model equation based on amplitude, midline, and frequency. Students do not choose the trigonometric function to model the situation. • F-TF.8: In Algebra II, Unit 11, Lesson 4 video, the teacher proves the Pythagorean Identity, but students do not prove the Pythagorean Identity. Students have multiple opportunities to find the sine, cosine, or tangent of an angle given the sine, cosine, or tangent and the quadrant of the angle. For example, in Algebra II, Unit 11, Lesson 10, students determine the values of $$\cos \theta$$ and $$\tan \theta$$ given $$\sin \theta=\frac{5}{13}$$ and the quadrant of the terminal ray. • G-CO.4: In Geometry, Unit 2, Lesson 3, students develop the definition of reflections by verifying a rigid motion between the preimage and image. Students draw segments connecting the preimage points with the corresponding points on the image and determine the angle the segments create with the line of reflection. Students use a compass to verify segments drawn between the point on the preimage and points on the image intersect the line of reflection at the midpoint. Students do not develop definitions of rotations and translations. • S-ID.2: In Algebra I, Unit 10, Lesson 3, students calculate the mean and median of data sets and determine which value is a better measure of the data. Students do not compare mean and median to the shape of the distribution. In Algebra I, Unit 10, Lesson 4, students calculate mean, interquartile range, and standard deviation of a data set. Within the same lesson, students compare standard deviations for the age of people who preferred two different sodas. Students do not compare standard deviation to the shape of the distribution. • S-ID.3: In Algebra I, Unit 10, Lesson 4, students compare two surveys of households with video-enabled devices. Students calculate the mean and standard deviation for each data set and determine the data set with the greatest variation. Students determine the number of households falling within one standard deviation of the mean. Students have limited opportunities to interpret differences in shape, center, and spread of data sets, accounting for possible effects of extreme data sets. • S-CP.4: In Algebra II, Unit 12, Lesson 3, students interpret a two-way frequency table of data sorted by gender and post-graduation plans and use the table to approximate conditional probabilities. In Algebra II, Unit 12, Lesson 5, students interpret a two-way frequency table of data sorted by eye color and hair color and use the sample space to decide if the events are independent. Students do not construct two-way frequency tables. ### Indicator 1c The materials require students to engage in mathematics at a level of sophistication appropriate to high school. 2/2 + - Indicator Rating Details The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics 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, apply key takeaways from grades 6-8, and vary the types of real numbers being used. Examples of applying key takeaways from grades 6-8 include: • In Algebra I, Unit 4, Lesson 1, students represent proportional relationships with equations between the total cost of apples and the number of apples bought (7.RP.2c). Within the same lesson, students graph an equation of a proportional relationship (7.RP.2d) and interpret the constant of proportionality represented in the graph. • In Algebra I, Unit 3, Lesson 7, students describe a function by identifying and interpreting the domain and range of a graph representing the height above ground and the time at a resort (8.F.5). Within the same lesson, students calculate and interpret the rate of change for two restricted intervals on the graph (8.F.4). • In Geometry, Unit 7, Lesson 12, students apply their knowledge of the Pythagorean Theorem (8.G.6,7) to prove the Pythagorean Theorem using similarity properties. • In Algebra I, Unit 10, Lessons 1-4, students apply an understanding of dot plots, box plots, and histograms (6.SP.4). • In Algebra II, Unit 10, Lessons 7, 8, and 10, students apply their knowledge of operations with rational numbers (7.NS.1,2) to add, subtract, multiply, and divide rational expressions. Examples of regularly using age-appropriate contexts include: • In Algebra I, Unit 6, Lesson 6, students calculate the amount of money a savings account contains after 5 years with an initial deposit of$450 and an annual interest rate of 3.5%.
• In Geometry, Unit 10, Unit 10 Review, students determine the volume of a shape created by a 3D printer. Students are also given the weight of the plastic in grams and asked to calculate the weight of the shape.
• In Algebra II, Unit 2, Lesson 1, students encounter a scenario about an internet music service where consumers pay a monthly rate of \$5 with an additional cost of 10 cents per song. Students determine the independent and dependent variables, write an equation modeling the function, and produce the graph over the interval 0x40.

Examples of the materials varying the types of numbers used include:

• In Algebra I, Unit 6, Lesson 3, students create equations for exponential growth and decay using integers, fractions, and decimals.
• In Algebra I, Unit 2, Lesson 13, students solve inequalities with rational number solutions.
• In Geometry, Unit 5, Lesson 6, students solve application problems using the Pythagorean Theorem. Triangle side lengths within the lesson are expressed as integers, and solutions include irrational numbers expressed as equivalent radical expressions.
• In Geometry, Unit 7, Lesson 2, students determine scale factor and create dilations with rational scale factors as well as side lengths expressed as radicals.
• In Algebra II, Unit 4, Lesson 1, students evaluate exponential functions with integer exponents, and in Lesson 2, students rewrite expressions with rational exponents as roots. In Algebra II, Unit 4, Lesson 6, students use exponential modeling with percent growth and decay. The lesson primarily contains integers to represent interest rates. The solutions contain integers and rational numbers, but the final solution is rounded to an integer.
• In Algebra II, Unit 8, Lesson 4, students simplify rational exponents.

### 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.
2/2
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Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series.

Examples of the instructional materials fostering coherence through meaningful mathematical connections in a single course include:

• In Algebra I, Unit 8, students solve quadratic equations using different methods. Methods include completing the square, factoring, and phase shifting. In Algebra I, Unit 9, Lessons 5 and 6, students calculate zeros by completing the square then extend to the quadratic formula (A-REI.4b, F-IF.8a).
• In Geometry, Unit 2, Lessons 1-3, students translate, rotate and reflect figures to uncover properties of rigid motion. In Geometry, Unit 5, Lessons 9-11, students continue to perform transformations on a coordinate plane (G-CO.5).
• In Algebra II, Unit 2, Lesson 6, students discover properties of inverse functions using tables and graphs of linear and quadratic functions. In Algebra II, Unit 4, Lesson 8, students examine logarithmic functions by creating a table and a graph of the inverse of $$y=2^x$$ (F-BF.4, F-IF.7e).

Examples of the instructional materials fostering coherence through meaningful mathematical connections between courses include:

• In Algebra I, Unit 8, Lesson 4, students identify the coordinates of the maximum or minimum of quadratic equations by completing the square to convert from standard form to vertex form. In Algebra I, Unit 9, Lesson 5, students complete the square to find zeros of a quadratic function (A-REI.4b). In Geometry, Unit 9, Lesson 10, students complete the square to write the equation of a circle in standard form and identify the radius and center of a circle (G-GPE.1). In Algebra II, Unit 6, Lesson 10, students also complete the square to determine the center and the radius of a circle (G-GPE.1).
• Students identify the effects of transforming graphs for several functions within the Algebra I and Algebra II materials. In Algebra I, Unit 8, Lesson 3, students perform horizontal and vertical shifts on quadratic functions. In Algebra I, Unit 9, Lesson 4, students perform horizontal and vertical shifts on square root functions and absolute value functions. In Algebra II, Unit 4, Lesson 9, students perform horizontal and vertical shifts on logarithmic functions. In Algebra II, Unit 7, Lesson 1, students perform horizontal and vertical shifts on linear functions, absolute value functions, piecewise functions, and quadratic functions. In Algebra II, Unit 8, Lesson 1, students perform horizontal and vertical shifts on square root functions. In Algebra I, Unit 8, Lesson 5, students perform vertical and horizontal stretching of quadratic functions. In Algebra I, Unit 11, Lessons 1- 4, students perform vertical and horizontal stretching of quadratic functions, piecewise functions, absolute value functions, exponential functions, and trigonometric functions. In Algebra II, Unit 7, Lesson 2, students perform reflections on square root functions (G-CO.5,6).

### Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
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Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The materials reference topics from Grades 6-8, but the instructional materials do not explicitly identify standards from Grades 6-8.

Examples where the materials reference topics from Grades 6-8 but do not explicitly identify content from Grades 6-8 include:

• In Algebra I, Unit 2, Lesson 2, the materials state that students have solved linear equations in 8th grade Common Core math.
• In Algebra I, Unit 4, Lesson 1, the materials noted proportional relationships being studied in previous courses.
• In Algebra I, Unit 4, Lesson 4, students graph linear equations in standard and slope-intercept form. The materials state, “Do you remember what this type of problem is called from 8th grade Common Core Mathematics?”.
• In Algebra I, Unit 5, Lesson 2, students solve systems of equations using substitution. The materials reference substitution as being in Common Core 8th grade mathematics.
• In Geometry, Unit 5, Lesson 6, students solve problems using the Pythagorean Theorem. The materials reference the Pythagorean Theorem relationship from middle school.
• In Geometry, Unit 10, Lesson 1, students calculate the perimeter of several types of figures. The materials state, “Since grade school you’ve learned that perimeter represents the distance or length of the path that surrounds a two-dimentional shape.”

Examples where connections between grades 6-8 and high school concepts are present and allow students to extend their previous knowledge:

• In Grade 8, students calculate and interpret the rate of change of a linear function from a description of a relationship, table, or graph (8.F.4). In Algebra I, Unit 3, Lesson 6, students calculate the average rate of change of linear and quadratic functions from a description of a relationship, tables, and graphs. In Algebra I, Unit 6, Lesson 4, students calculate the average rate of change over given intervals for exponential functions. In Algebra I, Unit 6, Lesson 8, students distinguish between situations that can be modeled with linear functions and with exponential functions by comparing the average rate of change for linear and exponential functions (F-LE.1).
• In Grade 8, students apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems (8.G.7). In Geometry, Unit 8, Lessons 4-6, students use the Pythagorean theorem and trigonometric ratios to determine unknown side lengths in right triangles in real-world and mathematical problems (G-SRT.9).
• In Grade 8, students construct scatter plots for bivariate measurement data and informally fit a straight line for scatter plots that suggest a linear association (8.SP.1-3). In Algebra I, Unit 10, Lessons 7-9, students use technology to fit a linear function to a scatterplot that suggests a linear association (S-ID.6a).
• In Grade 8, students know and apply the properties of integer exponents to generate equivalent numerical examples (8.EE.1). Students review the properties of integer exponents in Algebra I, Unit 6, Lessons 1 and 2. In Algebra II, Unit 4, Lesson 2, students extend their use of properties of exponents to rational exponents (N-RN.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.
0/0
+
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Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series do not explicitly identify the plus standards. Some plus standards are fully addressed by the materials and coherently support the mathematics which all students should study in order to be college and career ready. However, some plus standards are partially addressed by the materials, and some plus standards are not addressed by the materials.

The following plus standards are fully addressed in the series:

• F-BF.1c: In Algebra II, Unit 2, Lesson 3, and Unit 10, Lesson 5, students find values for a composition of functions.
• F-BF.4c: In Algebra II, Unit 2, Lesson 6, students read a graph to answer questions about the inverse function. Within the same lesson, students graph the inverse function given a table and a graph.
• F-BF.5: In Algebra II, Unit 4, Lesson 8, the materials define the relationship between exponents and logarithms. Within the same lesson, students evaluate logarithms using the relationship when needed.
• F-TF.3: In Algebra II, Unit 11, Lesson 3, students use special right triangles to determine geometrically the values of sine, cosine, and tangent, and in Lesson 5, students use the unit circle to express the values of sine, cosine, and tangent.
• G-C.4: In Geometry, Unit 9, Lesson 11, students construct a tangent line to a circle from a point outside the given circle.
• S-CP.8: In Algebra II, Unit 12, Lesson 6, students apply the general Multiplication Rule and determine if the outcomes are dependent or independent.

The following plus standard are partially addressed in the series:

• F-BF.4b: In Algebra II, Unit 2, Lesson 3, students perform function compositions. Students calculate g(f(15)), g(f(-3)), and g(f(x)). The students determine what is always true about the composition of the two functions f(x) = 2x + 9 and g(x) = $$\frac{x-9}{2}$$; however, students do not make the connection that the two given functions are inverses of each other. In Algebra II, Unit 2, Lesson 6, students perform function compositions to show inverses “undo” each other, yet students do not verify that one function is the inverse of another using composition.
• F-IF.7d: In Algebra II, Unit 10, Lesson 5, students graph a rational function and identify the type of symmetry. In Algebra II, Unit 10, Lesson 5, students identify the x-intercepts and y-intercepts of a rational function algebraically. Within the same lesson, students sketch the graph of a rational function and identify if it is an even or odd function. The materials do not identify asymptotes or show end behavior of a rational function.
• G-GMD.2: In Geometry, Unit 10, Lesson 8, the materials give an informal argument using Cavalieri’s principle for the volume of a prism. The materials do not give an informal argument using Cavalieri’s principle for the volume of other solid figures.
• S-MD.1: In Algebra II, Unit 12 Add-On, students graph a probability distribution. The materials do not define a random variable for quantity of interest.
• N-CN.3: In Algebra II, Unit 9, Lesson 2, students identify the conjugate of complex numbers. Students do not find moduli and quotients of complex numbers.
• N-CN.9: In Algebra II, Unit 10, Lesson 2, the materials reference the Fundamental Theorem of Algebra by having students generalize the relationship between the number of zeros and the highest power in a polynomial. Students make statements concerning the minimum and maximum number of zeros. The Fundamental Theorem of Algebra is not stated by name in the materials.
• A-APR.7: In Algebra II, Unit 1, Lesson 7, students multiply and divide rational expressions. In Algebra II, Unit 10, Lesson 8, students add and subtract rational expressions. In Algebra II, Unit 10, Lesson 9, students perform operations with complex fractions. The materials do not describe how rational expressions form a system analogous to the rational numbers.

The following plus standards are not addressed in the series:

• N-CN.4-6,8
• N-VM
• A-APR.5
• A-REI.8,9
• F-BF.4d
• F-TF.4,6-7,9
• G-SRT.9-11
• G-GPE.3
• S-CP.9
• S-MD.2-7

## Rigor & Mathematical Practices

### Criterion 2a - 2d

Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.
6/8
+
-
Criterion Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet the expectations for Rigor and Balance. The materials meet expectations for providing students opportunities in developing procedural skills and balancing the three aspects of Rigor, and the materials partially meet expectations for providing students opportunities in developing conceptual understanding and utilizing mathematical concepts and skills in applications.

### Indicator 2a

Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Across the series, conceptual understanding is developed in the instructional portion of the lessons as students are guided through the mathematical concepts by the teacher and/or instructional videos. The instructional videos explain all aspects of the concepts of the lessons, so the instructional materials do not provide opportunities for students to independently demonstrate conceptual understanding throughout the series.

Examples showing the development of conceptual understanding in the instructional portion of lessons and not allowing students to independently demonstrate their conceptual understanding include:

• In Algebra II, Unit 4, Lesson 2 instruction, students, with teacher and/or instructional video guidance, observe how the expression $$16^{\frac{1}{2}}$$ is equivalent to $$\sqrt{16}$$. Students then test this observation by evaluating $$25^{\frac{1}{2}}$$$$81^{\frac{1}{2}}$$, and $$100^{\frac{1}{2}}$$. The materials state the general relationship $$b^{\frac{1}{n}} =\sqrt[n]b$$, and students rewrite expressions with fractional exponents into expressions with radicals. The lesson concludes with materials stating the general relationship $$b^{\frac{m}{n}} =\sqrt[n]{b^m}$$, and students rewrite three expressions using this equivalence statement. In Lesson 2 Homework, students independently develop their fluency in rewriting and evaluating expressions involving rational exponents without having to explain how the definition of the meaning of rational exponents extends from the properties of integer exponents (N-RN.1).
• In Algebra I, Unit 8, Lesson 6 instruction, students, with teacher and/or instructional video guidance, identify the zeros of a graphed quadratic function, $$y=x^2-2x-3$$, and factor the quadratic expression, $$x^2-2x-3$$. The materials state the Zero Product Law, and the subsequent exercises use the Zero Product Law to find solutions of quadratic functions. While students encounter the relationship between zeros and functions of a quadratic, students do not investigate or explain the relationship on their own in this lesson. In Algebra I, Unit 8, Lesson 7 instruction, students, with teacher and/or instructional video guidance, find the zeros of a quadratic function algebraically and sketch a graph of the quadratic function. In Lesson 7 Homework, students consider a quadratic and cubic function and (i) find the zeros algebraically and (ii) sketch a graph of the function using their calculator. Students do not independently construct a graph using the zeros they calculated algebraically (A-APR.3).
• In Algebra II, Unit 9, Lesson 3 instruction, students encounter quadratic equations with complex solutions. Students, with teacher and/or instructional video guidance, consider the quadratic equation, $$y=x^2-6x+13$$, algebraically find the x-intercepts, and write the answers in equivalent complex forms. Students sketch a graph using their calculator before making an observation between the number of x-intercepts on the graphed parabola and the complex roots found algebraically. This is followed by an exercise in which students calculate the discriminant of several quadratic expressions to determine whether the given quadratic equations have x-intercepts. In Lesson 3 Homework, students use the discriminant to determine whether given quadratic equations have real or imaginary zeros (with no connection to graphing). In this lesson, students do not independently develop their conceptual understanding of recognizing when quadratic equations have complex solutions (A-REI.4b).
• In Algebra I, Unit 4, Lesson 10 instruction, the materials state “Any coordinate pair (x, y) that makes the equation or inequality true lies on the graph. Any coordinate pair (x, y) that makes the equation or inequality false does not lie on the graph.” In Lesson 10 Homework, students do not independently develop their understanding of this standard as they substitute (x, y) values to determine if a point lies on the graph of an equation (A-REI.10).

### Indicator 2b

Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meet expectations for providing opportunities for students to independently develop procedural skills, especially where called for in specific content standards or clusters. Across the series, students independently demonstrate procedural skills throughout the materials, specifically in the homework activities located after every lesson.

Examples showing independent development of procedural skills include:

• In Algebra II, Unit 8, Lesson 3, students transform exponential expressions into equivalent expressions using properties of exponents. In Algebra II, Unit 8, Lesson 4, students transform radical expressions to equivalent expressions with rational exponents and vice versa. In Algebra II, Unit 8, Lesson 5, students transform rational expressions into equivalent expressions (A-SSE.3c).
• In Algebra I, Unit 7, Lesson 1, students add and subtract polynomials. In Algebra I, Unit 7, Lesson 2, students multiply polynomials (A-APR.1).
• In Algebra I, Unit 8, Lesson 3, students transform parabolas to identify the effects on the corresponding quadratic function. In Algebra I, Unit 9, Lesson 3, students transform square root functions to identify the effects on the corresponding function. In Algebra I, Unit 11, Lesson 1, students identify characteristics of function transformations and transform functions. In Algebra I, Unit 11, Lesson 2, students perform horizontal and vertical stretches on functions. In Algebra II, Unit 7, Lesson 1, students transform functions through vertical and horizontal shifts. In Algebra II, Unit 7, Lesson 3, students transform functions through vertical stretching. In Algebra II, Unit 7, Lesson 4, students identify characteristics of horizontal stretching of a function (F-BF.3).
• In Geometry, Unit 5, Lesson 1, students use slope criteria to show lines are parallel. In Geometry, Unit 5, Lesson 2, students use slope criteria to show lines are perpendicular. In Geometry, Unit 5, Lesson 3, students draw a line parallel or perpendicular to a given line passing through a given point. Students then write the equation of the line drawn in slope-intercept form. In Geometry, Unit 5, Lesson 4, students apply the point-slope form of a line to create equations parallel or perpendicular to a given line (G-GPE.5).
• In Geometry, Unit 3, Lesson 4, students use congruence criteria for triangles to prove relationships in triangles. In Geometry, Unit 7, Lesson 6, students use similarity criteria for triangles to prove triangles similar. In Geometry, Unit 7, Lesson 7, students use similarity criteria for triangles to prove relationships in triangles (G-SRT.5).

### Indicator 2c

Attention to Applications: The materials support the intentional development of students' ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. The instructional materials do not routinely engage students in non-routine applications of mathematics throughout the series. The majority of the application problems are extensions of content requiring students to use skills introduced within the same lesson and not spanning unit or course topics.

Throughout the materials, students have limited opportunities to answer questions using numerous methods. The materials often instruct students to solve problems using a specific mathematical procedure, which makes the problems routine. Examples that show missed opportunities to engage in non-routine problems throughout the series include:

• In Algebra I, Unit 8, Lesson 6, students determine how long before a baking soda rocket fired upward hits the ground. Students use the given quadratic function $$h(t)=-16t^2+ 80t$$ and calculate the time algebraically by factoring. The scenario follows the lesson on factoring to calculate the zeros of a function, and the materials instruct students how to calculate solutions in one way. In Algebra II, Unit 8, Lesson 6, students use the given equation $$h=-9.8t^2+32.2t+6.5$$ that represents the height of a missile above the ground. Students do not choose a method for calculating the solution, instead the materials instruct students to use the quadratic formula to calculate when the rocket will hit the ground (A-REI.4).
• In Geometry, Unit 8, Lesson 1, students estimate the height of a ladder leaning against a wall. Students are given a model to represent the scenario and do not draw their own models to represent the scenario. The problem follows the lesson on similar, right triangles, and students use similar, right triangles to create a ratio to solve the problem (G-SRT.6).
• In Algebra II, Unit 6, Lesson 9, the materials provide a function, $$h(t)=-16t^2+80t+30$$, for the height of a tennis ball t seconds after it is thrown upwards. Students are instructed by the materials to determine the time of the tennis ball’s greatest height algebraically. Students are also directed to use a graphing calculator to sketch a graph of the ball’s height where $$t\geq0$$ and $$h\geq0$$. The materials then state for students to use the zero command on the calculator to calculate the time the ball stays in the air (F-IF.4). Students do not solve the scenario using different techniques since the materials provided instructions for calculating the solution.
• In Algebra II, Unit 3, Lesson 4, the materials provide a scenario of a rocket with 225 gallons of fuel taking off and using fuel at a constant rate of 12.5 gallons per minute. Students follow directions to create a linear model in the form, y = ax + b, where y is the amount of fuel and x is the number of minutes. The materials provide students with a general graph representing the scenario and instruct students to use a calculator to determine the intercepts of the model. The materials direct students to calculate the maximum number of minutes by graphing the horizontal line y = 50 to show the point of intersection (S-ID.C). The materials provide instruction for solving the problem which allows for one way of solving the scenario.

### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Procedural skills and fluency are present independently throughout the materials, and the materials develop conceptual understanding independently. Application problems are often connected with developing procedural skills and fluency throughout the materials.

Examples where all three aspects of rigor are present independently throughout the materials include:

• In Algebra I, Unit 7, Lesson 5 Homework, students build procedural skills by factoring trinomials. The materials separate the trinomials into three groups for students to build procedural skills gradually. In Algebra I, Unit 7, Lesson 6, students continue to build procedural skill by factoring a greatest common factor before factoring the trinomial completely.
• In Algebra I, Unit 8, Lesson 2 instruction, students are instructed to use a calculator to sketch $$y=2x^2$$, $$y=3x^2$$, $$y=4x^2$$, $$y=-2x^2$$, $$y=-2x^2$$, and $$y=-4x^2$$. After sketching the graphs, the materials provide statements about the direction a parabola opens that students must complete based on the quadratic equations graphed. Understanding of the relationship between parts of the equation of a parabola and how the parabola opens is developed.
• In Algebra II, Unit 8, Lesson 2 Homework, students build procedural skill in solving square root equations with one root.

Examples where multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic of study throughout the materials include:

• In Algebra I, Unit 5, Lesson 5, students solve a problem involving a local theater’s ticket prices for children and adults. Students write systems of equations to represent total tickets sold and the amount of profit. Students develop skill in solving systems of equations by calculating the number of child tickets sold and the number of adult tickets sold.
• In Geometry, Unit 9, Lesson 1, the materials provide terminology for students to conceptually understand relationships between parts of a circle. Within the Homework, students develop understanding by identifying parts of a circle and solving problems using characteristics of circles.
• In Algebra II, Unit 13, Lesson 1, the materials provide definitions of types of variability that can occur when collecting data. Students build conceptual understanding by providing real-world examples to represent each type of variability. Within the Homework, students apply understanding of the definitions to identify the type of variability present for several different real-world scenarios.

### Criterion 2e - 2h

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
4/8
+
-
Criterion Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for Practice-Content Connections. The materials do not identify the mathematical practices for teachers or students which results in the deduction of a point in indicator 2e. The materials intentionally develop the following mathematical practices to their full intent: make sense of problems and persevere in solving them (MP1), reason abstractly and quantitatively (MP2), model with mathematics (MP4), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8).

### Indicator 2e

The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.
0/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series do not meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. Throughout the series, the materials intentionally develop MP1 to the full intent of the practice standard. However, the materials do not intentionally develop MP6 to the full intent of the practice standard.

Examples that demonstrate the intentional development of MP1 to the full intent of the practice include:

• In Algebra I, Unit 2, Lesson 1, students complete a chart representing the percent of a body affected by a disease over a set amount of days. Students make sense of the data in the chart to provide an explanation to a possible patient about the disease and treatments.
• In Geometry, Unit 2, Lesson 3, students reflect a segment on a coordinate plane across the y-axis followed by a rotation of 90 degrees counterclockwise about the origin. Students determine the coordinates of the endpoints of the transformed segment. Students then use a ruler or tracing paper to determine whether the preimage and image are the same length. Finally, students consider whether their solution makes sense.
• In Algebra II, Unit 6, Lesson 3, students consider the trinomial $$6x^2-35x-6$$ and are given four possible factorizations. Students identify two factorizations that do not make sense and explain why they are unreasonable. Students then consider the two factorizations that are reasonable and determine which of the two are the correct factorizations.

Throughout the series, students are given few opportunities to choose variables when context is provided. The majority of the graphs in the series provide labels on the graph or provide instructions on how the graph is labeled. The materials frequently specify how students should round when calculating solutions, so students do not make sense of a degree of precision that is appropriate for any given context. Examples where the materials do not develop MP6 to the full intent of the practice include:

• In Algebra I, Unit 9, Lesson 6, the materials state, “Many times in applied problems it makes much greater sense to express answers, even if irrational, as approximated decimals.” This is followed by an exercise in which students are instructed to round their answer to the nearest tenth of a second. However, the materials do not provide guidance as to when it’s appropriate to round an answer.
• In Algebra II, Unit 3, Lesson 4, students write a linear model in point-slope and slope-intercept forms to represent the distance driven as a function of time. Materials identify the variables, D for distance and h for the number of hours, rather than students choose variables to represent the situation.
• In Algebra II, Unit 11, Lesson 9, the materials provide students the sinusoidal equation $$O(t)=11cos(\frac{\pi}{12}t)+71$$ representing the temperature of a summer day in New York. Students are instructed to graph the function given. Instead of allowing students to label the graph, the materials provide the graph with the y-axis representing the degrees and the x-axis representing the time in hours. The materials also provide the scale of the graph.
• In Geometry, Unit 10, Review, students find the combined area of fan blades to the nearest square centimeter in part (a) of the problem, calculate the volume of the blades to the nearest tenth of a cubic centimeter in part (b), and determine the density of the fan blades to the nearest hundredth of grams per cubic centimeter in part (c). The materials provide no rationale for why students express their answers to each of the different levels of precision for different parts within the same problem.

### Indicator 2f

The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. The materials intentionally develop MP2 to the full intent across the series, but the materials do not intentionally develop MP3 to the full intent across the series.

Examples that demonstrate the intentional development of MP2 across the series include:

• In Algebra I, Unit 10, Lesson 6, the materials provide a data set comparing the price people pay for their most expensive car and the current value of their house and a linear regression equation modeling the scenario. Students use the regression equation to quantitatively make predictions. Additionally, students reason abstractly about a possible causal relationship between the two variables. If students believe a causal relationship exists, then students identify which variable causes the other; whereas, if students do not believe a causal relationship exists, then students identify a third confounding variable.
• In Geometry, Unit 10, Lesson 10, the materials provide a model of a pill shaped like a cylinder with a hemisphere at each end. Students calculate the volume of the pill given the height of the cylinder as 12 mm and the height of the pill as 18 mm. Students determine the amount of Vitamin C a pill contains per cubic millimeter if the pill contains 100 milligrams of Vitamin C. Students compare the amount of Vitamin C in a second spherical pill with a diameter of 10 mm containing 1.5 milligrams of Vitamin C. In this problem, students attend to the meaning of quantities as they perform calculations with different units.
• In Algebra II, Unit 5, Lesson 2, students work with the Koch Snowflake and examine the perimeters of the first six iterations. The materials provide the perimeter of the first snowflake, and students quantitatively determine the perimeter of the second and sixth snowflakes. Students reason abstractly to explain why the perimeter would become infinitely large if the iteration process continued forever.

Throughout the series, the materials, and not students, make conjectures that students must examine or prove. Students do not recognize counter-examples when breaking down situations. While students justify or explain their reasoning throughout the series, students do not communicate their thinking to others or respond to arguments from other students. Students have limited opportunities to read the arguments of others and decide if they make sense, and students do not ask questions to clarify or improve the arguments.

Examples where the materials do not develop MP3 to the full intent of the practice include:

• In Algebra I, Unit 1, Lesson 6, students examine provided student work that represents a pattern. The materials tell students that the work is wrong, and students show why the work is wrong. Students find the correct pattern and explain their reasoning. The students do not independently determine if the student work is correct or incorrect.
• In Geometry, Unit 7, Lesson 8, students are given a model involving a triangle and parallel lines. Students explain how Francine incorrectly solved for a missing side length by using the Side Splitter Theorem, and students calculate the correct length. Students do not independently determine if the student work is correct or incorrect.
• In Algebra II, Unit 6, Lesson 5, the materials state, “Be careful when you use factoring by grouping. Don’t force the method when it does not apply. This can lead to errors.” After this, students explain the error in the factored expression of $$2x^3+10x^2+7x+21$$. Students do not independently determine if the work is correct or incorrect.

### Indicator 2g

The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series partially meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. The materials provide opportunities for students to model with mathematics (MP4), but MP5 is not developed to the full intent of the practice.

Examples of MP4 being used to enrich the mathematical content to the full intent of the practice include:

• In Algebra I, Unit 11, Lesson 4, students determine a linear function to model the volume of water in a tank over time. Students also determine an exponential function to model volume as a function of time. Lastly, students determine whether the exponential function or linear function is a better model for the problem.
• In Geometry, Unit 10, Lesson 5, students determine the weight of three propellor blades by calculating the area of each sector. While students are provided a diagram, radii, and angle measures, students create their own equations to solve the problem.
• In Algebra II, Unit 4, Lesson 6, students determine, “In how many years, to the nearest year, will Red Hook have a greater population than Rhinebeck?” Students create and graph exponential equations to determine their point of intersection.

The instructional materials often specify tools needed to solve problems, so students have limited opportunities to choose the appropriate tools to use for a scenario. Examples of the materials not developing MP5 to the full intent of the practice include:

• In Algebra I, Unit 8, Lesson 7, students find the zeroes of the polynomial equation, $$y=x^3+2x-8x$$. The materials instruct students to use a graphing calculator and also provide a graph indicating the maximum and minimum values of the x- and y-axes. Students do not choose the tool needed, and students do not choose how they use the tool to sketch the graph.
• In Algebra I, Unit 9, Lesson 7, students use the Quadratic Formula to determine which functions have real zeroes and which do not have real zeroes. The materials instruct students to use a graphing calculator using the standard viewing window to verify their solution. Students do not choose a tool to verify their solutions.
• In Geometry, the materials provide icons at the beginning of each lesson to indicate the tool needed to complete the lesson. For example, a ruler symbol would indicate a ruler is used in the lesson. In Geometry, Unit 1, Lesson 5, students create constructions to explain or verify the characteristics of circles. The materials include the compass, ruler, and protractor icons at the beginning of the lesson. Within the same lesson, the materials explicitly instruct students when to use a compass and a straight edge. Students do not choose a tool appropriate to the lesson.
• In Algebra II, Unit 1, Lesson 6, students solve expressions given an x-value. Students are instructed to use the STORE feature on the calculator to evaluate each expression. Additionally, the materials state, “The STORE feature is particularly helpful in checking to see if a value is a solution to an equation.” After the statement, students are instructed to solve the equation 6x-3=4x+9. Students use the STORE feature to determine if their solution is correct. Students do not choose the tool needed to verify their solution.

### Indicator 2h

The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for eMathInstruction Common Core for High School Mathematics series meets expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards.

Examples of MP7 being used to enrich the mathematical content to the full intent of the practice include:

• In Algebra I, Unit 2, Lesson 2, students consider the structure of equations in order to perform inverse operations. Within the same lesson, the materials instruct students to solve -2(x-4)+8=2 by reversing operations and using the distributive property first before solving equations. Students create equations from verbal expressions before solving.
• In Geometry, Unit 10, Assessment Form B, students consider a model for a plastic rocket that is composed of a cone and cylinder with a hemisphere hollowed out of the bottom of the cylinder. Students use structure to focus on the individual objects, rather than the entire model, in order to calculate the volume of the plastic that remains in the cylinder after it has been hollowed out.
• In Algebra II, Unit 10, Lesson 6, students consider the equivalent relationship between $$\frac{x^2- 4}{2x - 4}$$ and $$\frac{x + 2}2$$. Students rewrite the first expression to show the two expressions are equivalent. Then, students use a calculator to complete a table where they evaluate each expression for given values of x ranging from 0 to 4. Finally, students explain why the two expressions are not equivalent for x = 2.
• In Algebra II, Unit 11, Lesson 3, students explore patterns using the Pythagorean Theorem. The materials provide right triangles where students use the Pythagorean theorem to solve for missing values in radical form. Students use their discoveries to determine ordered pairs for different angles on the unit circle.

Examples of MP8 being used to enrich the mathematical content to the full intent of the practice include:

• In Algebra I, Unit 1, Lesson 7, the materials provide the steps for rewriting the expression, $$(2x^3)^3$$. Students write justifications to explain each of the steps, and students use repeated reasoning to write the extended products of $$(2^2)^4$$ and $$(x^3)^4$$ and write the extended products in the form of $$2^n$$ or $$x^n$$.
• In Geometry, Unit 3, Lesson 3, students prove triangles congruent using side-angle-side, angle-side-angle, and side-side-side criteria. Students attend to details to determine which congruent criteria will prove the triangles congruent. Within the same lesson, students choose a congruence theorem to justify why two triangles are congruent and explain their choice to evaluate the reasonableness of their solution.
• In Algebra II, Unit 5, Lesson 3, students find the sum for a variety of arithmetic sequences. Students summarize the results by determining the types of numbers used within the sequence in order to establish a rule for the sum of the arithmetic series.
• In Algebra II, Unit 10, Lesson 14, the materials state, “Squaring both sides of an equation is irreversible. Is cubing both sides of an equation reversible?”. Students answer the questions and provide numerical evidence to support their claim in order to evaluate the reasonableness of their solution.

## Usability

#### Not Rated

+
-
Gateway Three Details
This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two

### Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

### Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
N/A

### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
N/A

### Indicator 3c

There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.
N/A

### Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
N/A

### Indicator 3e

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
N/A

### Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

### Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
N/A

### Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
N/A

### Indicator 3h

Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.
N/A

### Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
N/A

### Indicator 3j

Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
N/A

### Indicator 3k

Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
N/A

### Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
N/A

### Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

### Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
N/A

### Indicator 3n

Materials provide support for teachers to identify and address common student errors and misconceptions.
N/A

### Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
N/A

### Indicator 3p

Materials offer ongoing assessments:
N/A

### Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
N/A

### Indicator 3p.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
N/A

### Indicator 3q

Materials encourage students to monitor their own progress.
N/A

### Criterion 3r - 3y

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

### Indicator 3r

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
N/A

### Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
N/A

### Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
N/A

### Indicator 3u

Materials provide support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
N/A

### Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.
N/A

### Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
N/A

### Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
N/A

### Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
N/A

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

### Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
N/A

### Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
N/A

### Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
N/A

### Indicator 3ac

Materials can be easily customized for individual learners.
N/A

### Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
N/A

### Indicator 3ac.ii

Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
N/A

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
N/A
abc123

Report Published Date: 05/07/2020

Report Edition: 2019

Title ISBN Edition Publisher Year
Common Core Algebra I - Spiral Bound 978-1-944719-02-9 eMath Instruction Inc. 2013
Common Core Algebra II - Spiral Bound 978-1-944719-06-7 eMath Instruction Inc. 2015
Common Core Geometry - Spiral Bound 978-1-944719-23-4 eMath Instruction Inc. 2018
Common Core Algebra II - Teacher Plus Answer Key Subscription / User (1 Year) Digital Edition eMath Instruction Inc. 2019

All publishers are invited to provide an orientation to the educator-led team that will be reviewing their materials. The review teams also can ask publishers clarifying questions about their programs throughout the review process.

Once a review is complete, publishers have the opportunity to post a 1,500-word response to the educator report and a 1,500-word document that includes any background information or research on the instructional materials.

Please note: Beginning in spring 2020, reports developed by EdReports.org will be using an updated version of our review tools. View draft versions of our revised review criteria here.

## Educator-Led Review Teams

Each report found on EdReports.org represents hundreds of hours of work by educator reviewers. Working in teams of 4-5, reviewers use educator-developed review tools, evidence guides, and key documents to thoroughly examine their sets of materials.

After receiving over 25 hours of training on the EdReports.org review tool and process, teams meet weekly over the course of several months to share evidence, come to consensus on scoring, and write the evidence that ultimately is shared on the website.

All team members look at every grade and indicator, ensuring that the entire team considers the program in full. The team lead and calibrator also meet in cross-team PLCs to ensure that the tool is being applied consistently among review teams. Final reports are the result of multiple educators analyzing every page, calibrating all findings, and reaching a unified conclusion.

## Rubric Design

The EdReports.org’s rubric supports a sequential review process through three gateways. These gateways reflect the importance of standards alignment to the fundamental design elements of the materials and considers other attributes of high-quality curriculum as recommended by educators.

• Materials must meet or partially meet expectations for the first set of indicators to move along the process. Gateways 1 and 2 focus on questions of alignment. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?
• Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom. In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

## Key Terms Used throughout Review Rubric and Reports

• Indicator Specific item that reviewers look for in materials.
• Criterion Combination of all of the individual indicators for a single focus area.
• Gateway Organizing feature of the evaluation rubric that combines criteria and prioritizes order for sequential review.
• Alignment Rating Degree to which materials meet expectations for alignment, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.
• Usability Degree to which materials are consistent with effective practices for use and design, teacher planning and learning, assessment, and differentiated instruction.

## Math HS Rubric and Evidence Guides

The High School review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

• Focus and Coherence

• Rigor and Mathematical Practices

• Instructional Supports and Usability

The High School Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways.

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom.

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.

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