Alignment: Overall Summary

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials meet the expectations for focus and coherence by being 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). In Gateway 2, the instructional materials meet the expectations for rigor and balance by reflecting the balances in the Standards and helping students meet the Standards' rigorous expectations, and the materials meet the expectations for mathematical practice-content connections by meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice.

See Rating Scale
Understanding Gateways

Alignment

|

Meets Expectations

Gateway 1:

Focus & Coherence

0
9
14
18
16
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

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

Usability

|

Meets Expectations

Not Rated

Gateway 3:

Usability

0
21
30
36
35
30-36
Meets Expectations
22-29
Partially Meets Expectations
0-21
Does Not Meet Expectations

Gateway One

Focus & Coherence

Meets Expectations

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).
16/18
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Criterion Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for focus and coherence. The instructional materials attend to the full intent of the mathematical content contained in the high school standards for all students, spend the majority of time on the CCSSM widely applicable as prerequisites, let students fully learn almost all non-plus standards, engage students in mathematics at a level of sophistication appropriate to high school, and make meaningful connections in a single course and throughout the series. The instructional materials partially attend to the full intent of the modeling process and partially identify and build on knowledge from Grades 6-8.

Indicator 1a

The materials focus on the high school standards.*
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.
4/4
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Indicator Rating Details

The instructional materials reviewed for Carnegie Learning Math Solutions Integrated series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials include few instances where all aspects of the non-plus standards are not addressed across the courses of the series.

The following are examples of standards that are fully addressed across courses in the series:

  • A-CED.3: In Math 1, Module 2, Topic 3, Activity 4.1, students determine constraints when writing inequalities to model the weight a raft can hold and the cost to ride a raft on a whitewater rafting trip. In Math 2, Module 3, Topic 1, Lesson 2, students determine constraints when writing equations and inequalities to model the range of acceptable weights for baseballs to be used at the professional level.
  • F-LE.1c: In Math 1, Module 3, Topic 2, Lesson 1, students recognize situations in which a quantity grows or decays by a constant percent rate per unit interval from a table, graph, equation, or problem context.
  • G-CO.12: In Math 1, Module 5 and Math 2, Modules 1 and 2, students make formal geometric constructions using a compass and straightedge and patty paper.
  • S-ID.6a: In Math 1, Module 1, Topic 3, Activity 2.3, students fit a linear function to represent the amount of antibiotic in a person’s body over time and assess whether the function is an appropriate fit for the data set. 

The following standards are not fully addressed across courses in the series:

  • A-REI.5: In Math 1, Module 2, Topic 3, Activity 2.3, students use linear combinations to solve a system of two equations in two variables within the context of price-saving specials offered at a resort. While students check their solution algebraically to confirm that linear combinations produces a correct solution for that particular system of equations, there was no evidence found where the materials or students justify their reasoning and prove that this method is true for other systems of equations.

Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.
1/2
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Indicator Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials do not use the full intent of the modeling process to address more than a few modeling standards across the courses of the series. 

The following examples use the full intent of the modeling process:

  • N-Q.2 and S-ID.6a: In Math 1, Module 3, Topic 2, Activity 4.1, students use data relating a driver’s Blood Alcohol Content (BAC) and the probability of a driver causing an accident to create a model that predicts the likelihood of a person causing an accident based on their BAC. While the problem is partially defined for the students, students formulate their own models (e.g., table, graph, and equation) and use those models to predict probabilities that drivers will cause an accident. Students interpret their findings to determine when a driver’s BAC is high enough to cause an accident and formulate guidelines around when it is safe for a person to drive, regardless of the legal BAC driving requirements. Students use their models to validate their guidelines as they engage in discussion with classmates over “safe to drive” vs. “legally able to drive.” In Activity 4.2, students report their findings in an article written for the newsletter of the local chapter of S.A.D.D. (Students Against Destructive Decisions).
  • G-GMD.3: In Math 3, Module 1, Topic 2, Lesson 5, students design planter boxes for windowsill store fronts, and certain requirements are provided regarding the materials available. Students complete a table of the height, width, length, and volume for different planter boxes and use their table to write a function to represent the volume of the planter box in terms of the height. After a worked example, students use a graph to validate their findings and determine possible heights for a planter box with a given dimension. Students report their findings when they contact a customer, who is seeking a planter box with a given volume, with possible dimensions.
  • S-IC.1-6: In Math 3, Module 5, Topic 2, Activity 1.3, students design and implement a plan to find out how much time teens, ages 16-18, spend online daily. Students select a data collection method and formulate questions. In Activity 2.3, students select a sampling method and conduct their survey. In Activity 3.4, students calculate the sample mean and the sample standard deviation of their data and use this information to determine the 95% confidence interval for the range of values for the time teenagers, ages 16-18, spend online each day. In Activity 4.6, students apply their calculations from Activity 3.4 and use statistical significance to make inferences about the population based on their collected data. In Activity 5.1, students report the results by writing a conclusion that answers their question of interest using their data analysis to justify the conclusion. The modeling process is scaffolded for the students through the five activities.

All aspects of the modeling cycle are addressed throughout the series, however, students are often limited in their opportunities to make choices and assumptions when defining mathematical modeling problems and validating their conclusions and improving upon their models when appropriate. Examples of how students engage in some, but not all, elements of the modeling process include:

  • N-Q.1: In Math 1, Module 1, Topic 1, Lesson 1, Talk the Talk, students write a scenario and sketch a graph to describe a possible trip to school. Students identify variables in the situation and determine how to represent key “points” on their possible trip to school in the graph. Students interpret their graph by making explicit connections to their scenario. When sharing their scenario and graph, students have the option to validate their model after observing similarities and differences between classmates’ scenarios and graphs. Students do not perform computations using their scenarios and models.
  • A-SSE.4: In Math 3, Module 3, Topic 4, Activity 1.3, students calculate credit card monthly payments when considering paying the minimum balance each month for a $1000 purchase to a credit card with 19% interest. The activity directs students to calculate the remaining balance after paying off the minimum balance for each month for a year. Students generalize their findings by writing a formula for the geometric series that represents the total monthly payment, the total payment toward principal over time, and the total payment toward interest over time. Students validate their model and perform computations as they consider other minimum monthly payments. In Talk the Talk after Activity 1.3, students consider what to look for when applying for a credit card and develop a plan to pay off their bills using their findings from the activity. Students do not define the problem.
  • F-BF.1: In Math 3, Module 1, Topic 2, Activity 3.1, students examine a civil engineer’s design for a storm drainage system. Students use a sheet of paper to model a drain and consider what measurements need to be calculated in order to allow the greatest flow of water through the drain. An equation and graph are formulated to show the relationship between the cross-sectional area of the drain and the height of the drain. Students analyze and interpret features of the graph and seek to find the maximum cross-sectional area for the drain pipe. Students do not perform any computations or validate and report their findings.
  • G-SRT.8: In Math 2, Module 2, Topic 2, Activity 2.2, students use the vertical rise and angle of elevation for a proposal for a wheelchair ramp. Students identify what information is required to show the ramp meets safety guidelines and calculate tan 4° to determine whether the ramp meets safety rules. Students do not formulate a model on their own or validate and report their findings.

In the materials, many lessons are structured with learning opportunities which contain step-by-step instructions for students with minimal opportunities for creativity, estimation, and student choice of math concepts and skills to combine and utilize for problem solving. At the end of topics and/or lessons, Performance Tasks can be found in assessment sections. The full modeling process is present within these tasks; however, the tasks are found within a summary assignment of scaffolded lessons which directs how students should mathematize the problem along with predictions and analyses that should occur.

Indicator 1b

The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
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
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Indicator Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers (WAPs) when used as designed. Examples of how the materials spend the majority of the time on the WAPs include:

  • N-RN.2: In Math 1, Module 3, Topic 3, Activity 1.4 and Math 3, Module 3, Topic 1, Lesson 4, students rewrite expressions involving radicals and rational exponents.
  • A-SSE.1a: In Math 1, Module 2, Topic 1, Activity 2.3, students consider linear expressions in general and factored form, and describe the contextual and mathematical meaning of each part of the equivalent expressions. In Math 2, Module 3, Topic 3, Activity 2.2, students identify the leading coefficients and y-intercepts from factored form and general form equivalent quadratic functions.
  • F-IF.4: In Math 1, Module 2, Topic 1, Activity 2.2, students analyze a linear graph relating the potential earnings based on the number of t-shirts sold at a festival. Students interpret the meaning of the origin, identify and interpret the slope, identify and interpret the x- and y-intercepts, and identify and interpret a feasible domain and range. In Math 1, Module 3, Topic 2, Activity 1.2, students sketch an exponential growth and exponential decay graph given a verbal description of two town populations. Students analyze and interpret the y-intercepts of each function and make a connection between the y-intercept and the equation of the exponential function. In Math 2, Module 3, Topic 1, Activity 3.2, students describe a possible scenario to model a piecewise graph showing the charge remaining on a cell phone battery over time and then determine the slope, x-intercepts, and y-intercepts and describe what each means in terms of the problem context. In Math 2, Module 3, Topic 3, Activity 1.3, students interpret the maximum or minimum, y-intercept, and x-intercept within the context of a pumpkin being released from a catapult.
  • G-SRT.5: In Math 1, Module 5, Topic 3, Activities 3.2 and 3.3; Math 2, Module 1, Topic 3, Activity 3.1; and Math 2, Module 2, Topic 1, Lesson 4, students use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures.
  • S-ID.7: In Math 1, Module 1, Topic 3, Lessons 1 and 2, students interpret linear models by graphing data on a scatter plot, determine an equation for a line of best fit, interpret the slope and intercept within the context of the data, and compute and interpret the correlation coefficient.

Indicator 1b.ii

The materials, when used as designed, allow students to fully learn each standard.
4/4
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Indicator Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for, when used as designed, letting students fully learn each non-plus standard. The following non-plus standards would not be fully learned by students:

  • A-SSE.4: In Math 3, Module 3, Topic 4, Activity 1.1, students do not derive the formula for a geometric series. An example is provided and students analyze the example to find a pattern in one question with two parts. Underneath the question, the materials give the formula to compute any geometric series. Students use the geometric series to solve problems. 
  • A-REI.4a: In Math 2, Module 4, Topic 1, Activity 5.1, the materials derive the quadratic formula by completing the square. Students do not derive the quadratic formula on their own.
  • A-REI.11: Students have limited opportunities to explain why the x-coordinates of the points where the graphs of two equations intersect are solutions. In Math 1, Module 2, Topic 3, Lesson 1, students find the intersection of two linear equations and explain why the x- and y-coordinates of the points where the graphs of a system intersect are solutions. In Math 2, Module 3, Topic 3, Activity 1.3, students find the intersection of linear and quadratic equations and explain why the x- and y-coordinates of the points where the graphs intersect are solutions. In Math 2, Module 4, Topic 2, Lesson 3, students find the solutions to systems of quadratic equations. Students do not explain this relationship for absolute value, rational, exponential, and logarithmic functions.
  • G-C.5: In Math 2, Module 2, Topic 3, Lesson 2 Getting Started, students use a dartboard of 20 sectors to determine the area of the entire dartboard and the area of one sector. Then students find the area of one sector if the dartboard was divided into 40 sectors. Students do not generalize their findings to a dartboard with n sectors and are given the formula for the area of a sector to begin Activity 2.1.

Indicator 1c

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
2/2
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Indicator Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from Grades 6-8.

Examples where the materials illustrate age appropriate real-world contexts for high school students include:

  • In Math 1, Module 3, Topic 1, Activity 1.3, students identify an exponential function to model a healthy breakfast challenge in which four students take selfies of themselves eating a healthy breakfast and send their selfies to four friends challenging them to do the same the next day and for four continuous days.
  • In Math 2, Module 2, Topic 1, Activity 1.1, students use dilations and scale factors within the context of zooming in and out on a tablet.
  • In Math 2, Module 4, Topic 2, Lesson 2, Getting Started, students model the path of a firework using a quadratic function.
  • In Math 3, Module 2, Topic 1, Lesson 4, Getting Started, students consider a polynomial function that represents the profit model for a landscaping company over time and consider what increasing and decreasing intervals represent within the context of the scenario.
  • In Math 3, Module 2, Topic 3, Activity 6.1, students use a rational function to determine the time needed for two teams to work together on attaching advertisements to the boards in a hockey rink.

Examples where students apply key takeaways from Grades 6-8 include:

  • In Math 1, Module 2, Topic 1, Activities 2.2, 2.3, and 2.4, students extend their Grade 8 knowledge of functions to interpret important features of a graph of a linear function and transformations of the original linear function.
  • In Math 1, Module 2, Topic 4, Activity 3.5, students apply their knowledge of area to approximate the area of France, using a map superimposed on a coordinate plane, and approximate the population when given the population density of the country.
  • In Math 2, Module 2, Topic 2, students apply their knowledge of ratios to develop their understanding of the trigonometric ratios of tangent, sine, and cosine.
  • In Math 2, Module 5, Topic 1, students apply their knowledge of probability to determine the probability of independent events and dependent events, as well as problems involving conditional probability.
  • In Math 3, Module 3, Topic 1, Activities 5.1 and 5.2, students expand upon their knowledge of square roots and cube roots to solve rational equations. 
  • In Math 3, Module 4, Topic 1, Activity 3.2, students apply their knowledge of unit conversions to convert between radians and degrees as units of measures to describe angles.

The materials primarily use integer values in examples, problems, and solutions in Math 1 and expand to other types of real numbers in Math 2 and Math 3. Students use radicals in certain content areas (e.g., Pythagorean Theorem, trigonometric functions, and quadratic formula). Examples where the materials include various types of real numbers include:

  • In Math 1, Module 2, Topic 2, Activity 2.3, students rewrite the formulas for surface area and volume of a cylinder for height and substitute decimal values for the radius, surface area, and volume to determine the height.
  • In Math 2, Module 2, Topic 1, Lesson 5, students use similarity of triangles to solve for unknown measurements when given measurements are expressed as integers, decimals, or square roots.
  • In Math 2, Module 5, Topic 2, Activity 5.1, students calculate geometric probability of throwing a dart in a shaded region of several different dartboards. Final probabilities are expressed as decimals or irrational numbers.
  • In Math 3, Module 1, Topic 2, Activity 3.2, students design a new town drainage system and describe the drain that has the maximum cross-sectional area for a piece of sheet metal that is 15.25 feet wide.
  • In Math 3, Module 2, Topic 1, Activity 4.3, students answer questions using the polynomial equation, $$b(t) = 0.000139x^4 - 0.0188x^3 + 0.8379x^2 - 13.55x + 176.51$$, which models a person’s glucose level.

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 the Carnegie Learning Math Solutions Integrated 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 Math 1, Module 1, Topic 2, Activity 2.1, students analyze patterns in sequences and then formally identify sequences as arithmetic or geometric. In Module 1, Topic 2, Activity 2.2, students match sequences to their appropriate graphs and verify that all sequences are functions. In Module 2, Topic 1, Activity 1.1, students use their knowledge of arithmetic sequences to write a linear function in the form f(x) = ax + b, making an explicit connection between the common difference of an arithmetic sequence and the slope of a linear function. (F-BF.1)
  • In Math 2, Module 4, Topic 1, Activity 4.4, students complete the square to determine the roots of a quadratic equation. In Activity 4.5, students rewrite a quadratic equation to identify the axis of symmetry and the vertex. In Activity 5.1, students derive the quadratic formula. In Topic 3, Activity 1.2, students write the general equation of a circle. (A-SSE.3a,b)
  • In Math 3, Module 1, Topic 2, Activity 4.1, students build a cubic function from a quadratic and linear function. In Module 1, Topic 2, Activity 6.2, students decompose a cubic function into three linear functions. In Module 1, Topic 3, students graph and analyze key characteristics of polynomials functions. (F-IF.7c) Students also use their knowledge of the characteristics of polynomial graphs to determine a polynomial regression model and use the regression model to make predictions (S-ID.6a).

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

  • In Math 1, Module 2, Topic 4, Activity 1.4, students classify a quadrilateral on a coordinate plane by calculating the length and slope of each line segment in the quadrilateral. In Math 2, Module 1, Topic 1, Activity 2.3, students generalize relationships about sides, angles, and diagonals for all quadrilaterals after investigating certain relationships with a ruler, protractor, and patty paper. In Math 2, Module 1, Topic 3, Lesson 2, students prove many of the relationships involving sides, angles, and diagonals in quadrilaterals from conjectures made earlier in the module (G-CO.11).
  • Materials include a Remember thought bubble to reinforce the definition of sine first introduced in Math 2, Module 2, Topic 2, Lesson 3, which is later used in Math 3, Module 4, Topic 1, Activity 1.1 to derive the formula A=$$\frac{1}{2}$$ab sin(C) for the area of a triangle.
  • Students identify the effect on the graph of f(x) when it is replaced by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k when transforming functions throughout the series (F-BF.3). In Math 1, students transform linear functions in Module 2, Topic 1, Lesson 3, and exponential functions in Module 3, Topic 1, Lesson 3. In Math 2, students transform absolute value functions in Module 3, Topic 1, Lesson 1, and quadratic functions in Module 3, Topic 3, Lesson 3. In Math 3, students transform polynomial functions in Module 1, Topic 3, Lesson 2, rational functions in Module 2, Topic 3, Lesson 2, radical functions in Module 3, Topic 1, Lesson 3, exponential and logarithmic functions in Module 3, Topic 2, Lesson 4, and trigonometric functions in Module 4, Topic 1, Lessons 5 and 6.

Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
1/2
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Indicator Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials do not explicitly identify content from Grades 6-8. However, each topic within each module across the series begins with a Family Guide that discusses “Where have we been?” in which connections to middle school content is addressed yet not connected to specific standards, and “Where are we going?” that outlines learning goals aligned to high school standards for the topic.

Examples where the materials make connections between Grades 6-8 and high school content and build on previous knowledge include:

  • In the Family Guide for Math 1, Module 2, Topic 1, the materials state, “Throughout middle school, students have had extensive experience with linear relationships. They have represented relationships using tables, graphs, and equations. They understand slope as a unit rate of change and as the steepness and direction of a graph.” In Topic 1, students extend their knowledge of linear functions as they transform linear functions and see how equations and graphs are affected by specific transformations.
  • In the Module Overview for Math 1, Module 5, the materials state, “In grade 8, students verified experimentally the properties of rotations, reflections, and translations and developed an understanding that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rigid motion transformations.” In the module, students use their knowledge of transformations to define each rigid motion transformation. 
  • In the Topic Overview for Math 2, Module 1, Topic 2, the materials state, “In elementary and middle school, students investigated lines, angles, triangles, and quadrilaterals. They used informal arguments to establish facts about the angle sum and exterior angles of triangles, as well as the angles created when parallel lines are cut by a transversal.” In Lesson 1, students use prior knowledge related to angles and lines to apply properties to angle measures, line segments, and distances.
  • In the Family Guide for Math 2, Module 4, Topic 3, the materials state, “Students first learned the Pythagorean Theorem in middle school. They have used it to solve for distances on the coordinate plane…” In Activity 2.4, students use the Pythagorean Theorem or the Distance Formula to determine given points that lie on a circle centered on the origin within the context of a video game.
  • In the Family Guide for Math 3, Module 2, Topic 3, the materials state, “Students have been working with rational numbers since elementary school.” In Lesson 1, students encounter rational functions as they graph reciprocals of $$f(x) = x$$ and $$g(x) = x^2$$ and generalize their findings to graphs of all reciprocal power functions. 

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 the Carnegie Learning Math Solutions Integrated series explicitly identify the plus standards and use the plus standards to coherently support the mathematics which all students should study in order to be college and career ready. All plus standards are explicitly identified in the standards overview chart at the beginning of each course. No plus standards are addressed in Math 1. When plus standards are addressed in Math 2 and Math 3, they are generally included in the last lesson of a topic as a purposeful extension of course-level work. When included, plus standards do not distract from learning the non-plus standards and can be omitted without impacting instruction and student learning.

The plus standards that are fully addressed include:

  • N-CN.8: In Math 2, Module 4, Topic 2, Activity 1.5, students work with the polynomial identity for the sum of two squares.
  • N-CN.9: In Math 2, Module 4, Topic 2, Activity 1.5, Talk the Talk, students explore the Fundamental Theorem of Algebra by completing a table for different quadratic equations and determining it is true for all quadratic polynomials when answering a “Who’s correct?” question. In Math 3, Module 1, Topic 2, Activity 6.3, students connect a graph to the Fundamental Theorem of Algebra for quadratic and cubic functions.
  • A-APR.5: In Math 3, Module 2, Topic 2, Activity 2.1, students use Pascal’s Triangle to expand binomials and generalize their findings when using the Binomial Theorem in Activity 2.2.
  • A-APR.7: In Math 3, Module 2, Topic 3, Lesson 4, students add, subtract, multiply, and divide rational expressions. In Activities 4.1, 4.2, 4.4, and 4.5, the materials include worked examples of how operations with rational numbers are similar to operations with rational expressions involving variables. In Activity 4.5, Talk the Talk, students determine whether the set of rational expressions is closed under the four operations.
  • F-IF.7d: In Math 3, Module 2, Topic 3, Activity 1.1, students graph rational functions and identify the end behavior and asymptotes of $$f(x) = 1/x$$. In Activity 1.2, students graph and identify the end behavior and asymptotes of $$f(x) = 1/x^2$$ and the reciprocals of all power functions, focusing on the key characteristics of the graphs between the reciprocals of the even power functions and the reciprocals of the odd power functions. In Module 2, Topic 3, Lesson 2, students graph transformed rational functions and identify zeros and asymptotes when factorizations are available.
  • F-BF.4b: In Math 3, Module 3, Topic 1, Activity 2.4, students compose functions to show that $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$$ are inverses of each other for x ≥ 0. Students compose functions within the context of verifying that two functions are inverses of each other.
  • F-BF.4c: In Math 3, Module 3, Topic 1, Activity 1.1, students use patty paper to “switch” the axes for $$L(x) = x, Q(x) = x^2, and C(x) = x^3$$. In Activity 2.1, students use ordered pairs of $$f(x) = x^2$$ in a table and “what (they) know about inverses” to graph the inverse $$y = ±\sqrt x$$. 
  • F-BF.4d: In Math 3, Module 3, Topic 1, Activities 2.1 and 2.2, students restrict the domain to produce an invertible function from a non-invertible function.
  • F-BF.5: In Math 3, Module 3, Topic 2, Lesson 3, students learn about the inverse relationship between exponents and logarithms through an explicit connection between the key characteristics of a graph for an exponential function and logarithmic function. In Topic 3, Lessons 3, 4, and 5, students use this inverse relationship to solve problems involving logarithms and exponents.
  • F-TF.3: In Math 3, Module 4, Topic 1, Activity 4.3, Talk the Talk, students identify the values of sine and cosine for π/3, π/4 and π/6. Students identify the values of tangent for π/3, π/4 and π/6 in the unit circle in Activity 6.3 Talk the Talk. In Lesson 6, students have the opportunity to use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number.
  • F-TF.4: In Math 3, Module 4, Topic 1, Activity 6.3, Talk the Talk, students use symmetry to determine the values of trigonometric functions at certain input values. In Topic 1, Activity 4.3, students learn the periodicity identity for sine and cosine functions and explore the periodicity of tangent functions in Activity 6.1.
  • G-SRT.9: In Math 3, Module 4, Topic 1, Activity 1.1, students derive the formula A = 1/2ab sin(C) for the area of a triangle.
  • G-SRT.11: In Math 3, Module 4, Topic 1, Activity 1.4, students apply the Law of Sines and the Law of Cosines to find unknown measurements in triangles in real-world contexts of surveying distances and flight paths.
  • G-C.4: In Math 2, Module 1, Topic 2, Activity 5.4, the materials provide step-by-step instructions for how to construct tangent lines to a circle through a point outside of the circle.
  • S-CP.8: In Math 2, Module 5, Topic 1, Activity 2.3, students apply the general multiplication rule to solve probability problems involving dependent events.
  • S-CP.9: In Math 2, Module 5, Topic 2, Lesson 3, students use permutations and combinations to compute probabilities of compound events and solve problems.
  • S-MD.5, S-MD.6: In Math 2, Module 5, Topic 2, Lesson 5, students are given $200 and either keep their money or return their money and spin a wheel to determine their winnings. Students explore probabilities of spinning the wheel and expected values in order to make a fair decision.
  • S-MD.7: In Math 2, Module 5, Topic 2, Lesson 5, Getting Started, students solve the Monty Hall Problem. In the problem, a student is on a game show and has to choose 1 of 10 doors they think a car is behind. The student picks one door, the game show host reveals 8 other doors that does not have the car behind them, and now the student has to decide to stick with their original door or switch to the only other door remaining. Students analyze the probability of the decision to keep or trade their door using probability concepts from the module.

The plus standards that are partially addressed include:

  • F-BF.1c: In Math 3, Module 3, Topic 1, Lesson 2, students use composition of functions to determine whether two functions are inverses of each other. Students do not use composition of functions in application problems.
  • G-SRT.10: In Math 3, Module 4, Topic 1, Activity 1.2, students derive the Law of Sines. In Activity 1.3, students derive the Law of Cosines. In both activities, students complete provided steps for the derivation of the trigonometric laws. In Activity 1.4, students use the trigonometric laws to solve problems. 
  • G-GMD.2: In Math 2, Module 2, Topic 3, Activity 4.2, students use Cavalieri’s Principle to understand the formulas for the volume of a cone and pyramid. Students do not use Cavalieri’s Principle for the volume of a sphere, rather, the materials simply state the formula in Activity 4.5.

The following plus standards are not addressed in the series:

  • N-CN.3-6
  • N-VM
  • A-REI.8,9
  • F-TF.6,7,9
  • G-GPE.3
  • S-MD.1-4

Gateway Two

Rigor & Mathematical Practices

Meets Expectations

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.
8/8
+
-
Criterion Rating Details

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet expectations for 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.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, the instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding.

Examples that show the development of conceptual understanding throughout the series include:

  • A-SSE.1b: In Math 2, Module 4, Topic 1, Activity 5.2, students interpret how the first term of each root obtained from the quadratic formula is represented graphically. Students then consider why the second term of each root obtained from the quadratic formula is the same except for the sign and how this is represented graphically. At the conclusion of the activity, students generalize their findings by labeling the vertex, axis of symmetry, roots, and distance each root lies from the axis of symmetry for quadratics of the form $$y = ax^2 + bx + c$$ for a > 0 and a < 0 with two real roots and double real roots.
  • F-IF.1: In Math 1, Module 1, Topic 1, Lesson 3, students develop their understanding of functions. In Activity 3.1, students analyze relations represented as an ordered pair, written description, graph, table, mapping, and equation to determine whether the relations are functions. In Activity 3.2, students identify the domain and range of functions by writing the domain and range in words or using inequality notation.
  • G-CO.7: In Math 1, Module 5, Topic 3, Activity 2.1, students use a worked example to explain why a segment can be mapped onto itself in at most two reflections. In Activity 2.2, students use that criteria to demonstrate two triangles are congruent using the SSS congruence theorem.

Examples that show the materials providing an opportunity for students to independently demonstrate conceptual understanding throughout the series include:

  • A-APR.2: In Math 3, Module 2, Topic 1, Activity 2.3, students divide polynomials and observe relationships between factors, divisors, dividends, and remainders to develop their understanding of the Remainder Theorem.
  • G-SRT.6: In Math 2, Module 2, Topic 2, Activities 1.1 and 1.2, students use properties of similar right triangles to compare side length ratios for 45-45-90 and 30-60-90 triangles. These activities set the foundation for students to develop the definitions of trigonometric ratios later in the topic.
  • S-ID.3: In Math 1, Module 4, Topic 1, Lesson 2 Assignment, students create a data set of 15 numbers where the mean and median are both 59 and the standard deviation is between 10 and 11. Students add an outlier to the data set and explain how the center and spread are affected.

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 for the Carnegie Learning Math Solutions Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Opportunities for students to independently demonstrate procedural skills across the series are provided in activities and assignments in the student materials, the online skills practice workbook, and MATHia software. 

Examples that show the development of procedural skills across the series include:

  • A-SSE.1b: In Math 2, Module 3, Topic 2, Activities 2.2 and 2.3, students interpret a and b in exponential functions of the form $$f(x) = a(b^x)$$ within the context of population growth and decay. There are several additional practice problems in the Math 2, Module 3, Topic 2 skills practice workbook.
  • A-SSE.2: In Math 3, Module 2, Topic 3, students rewrite rational expressions when graphing rational functions to determine asymptotes or discontinuities and when performing operations with rational expressions. Procedural skill practice is included in activities and assignments throughout the topic as well as in the Math 3, Module 2, Topic 3 skills practice workbook and MATHia.
  • F-IF.7b: In Math 2, Module 3, Topic 1, Lessons 3 and 4, students graph piecewise-defined functions, including absolute value and step functions. In Math 3, Module 3, Topic 1, Activities 2.1 and 2.2, students graph square root functions. In Math 3, Module 3, Topic 1, Activity 2.3, students graph square root and cube root functions. The skills practice workbooks provide additional opportunities for students to independently demonstrate their procedural skills in graphing these functions.
  • G-GPE.4: In Math 1, Module 2, Topic 4, Lesson 1, students use the distance formula and the slope criteria for parallel and perpendicular lines to classify triangles and quadrilaterals on a coordinate plane. In Math 2, Module 4, Topic 3, Lesson 2, students determine whether a given point lies on a circle on the coordinate plane. There are additional problems in the Math 1, Module 2, Topic 4, and Math 2, Module 4, Topic 3 skills practice workbooks.
  • G-SRT.5: In Math 2, Module 2, Topic 1, Lesson 5, students solve problems using triangle similarity. Both MATHia and the skills practice workbook provide additional opportunities for students to independently demonstrate their procedural skills in using triangle similarity to solve problems.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that 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. The instructional materials include multiple opportunities to engage students in routine and non-routine applications of mathematics throughout the series. Applications are included in single activities in each course and span several lessons and topics throughout a module. Performance task assessments corresponding to each topic often engage students in an application of mathematics in a real-world context.

Examples where students engage in the application of mathematics throughout the series include:

  • N-Q.2: In MATHia, Math 1, Searching for Patterns, Identifying Quantities Workspace, students watch an animation of a skier and decide on appropriate quantities to model the scenario.
  • A-CED.1: In Math 1, Module 2, Topic 2, Activity 3.1, students write a linear equation to represent total sales as a function of the number of boxes of popcorn sold for a fundraiser. In Topic 2, Lesson 3 assignment, students write a linear equation to represent the cost of a trip as a function of the number of gallons of gas for the trip. In Topic 2, Lesson 4 assignment, students write an inequality to represent the cost to produce t-shirts in a month making a profit of at least $2,000 but no more than $10,000.
  • F-IF.4: In Math 2, Module 3, Topic 1, Activity 3.2, students interpret key features of a graph of a piecewise function that models the percent of charge remaining on a cell phone battery over time. Students write a possible scenario that models the graph and explain what the slope, x-intercepts(s), and y-intercept represent in terms of the problem context. Students also write a scenario to model their own cell phone use during a typical day, graph the scenario, and determine the equation of the piecewise function.
  • G-SRT.8: In the Math 2, Module 2, Topic 2, Performance Task, students use trigonometric ratios to compare the height of two drones, determine the distance between the two people controlling the drones, and calculate the angle of elevation from one person to their flying drone.
  • S-ID.4: In Math 3, Module 5, Topic 1, Getting Started and Activity 3.1, students recognize that a data set representing the fuel efficiency for a sample of hybrid cars is normally distributed. Students use the Empirical Rule and a table to estimate areas under the normal curve.

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 for the Carnegie Learning Math Solutions Integrated series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding. Each topic includes: activities and assignments that develop students’ conceptual understanding and procedural skills in the student materials, skills practice worksheet that allows students additional practice to develop procedural skills, and performance tasks that assess students’ conceptual understanding and/or procedural skills often times in the context of a real-world scenario. 

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

  • In Math 3, Module 2, Topic 1, Activity 1.2, students use chunking as a method to factor quadratics that have common factors in some of the terms, but not all the terms. Students use their procedural knowledge of chunking to identify and factor perfect square trinomials. In Module 2, Topic 1, Activity 2.2, students use their procedural knowledge of chunking to rewrite the difference of two cubes and build their conceptual understanding when determining the formula for factoring the difference of two cubes. 
  • In Math 2, Module 4, Topic 2, Performance Task, students examine a table relating ticket prices and number of tickets sold for a baseball game, and students determine the linear regression equation that models the data. Students add a column to the table for the total amount of money earned from ticket sales and determine the quadratic equation that models the data. Students identify key features of the two graphs, describe these key features in the context of the problem, and explain why the data sets are modeled by different functions.
  • In Math 1, Module 1, Topic 2, Performance Task, the materials provide two scenarios (written description and diagram) for how to track the growth of a tree. Students recognize that one scenario represents an arithmetic sequence, whereas the other scenario represents a geometric sequence. Students represent the two sequences using a table, graph, and equation. Students’ conceptual understanding of the differences between arithmetic and geometric sequences and their procedural knowledge of how to write recursive and explicit formulas are assessed in this performance task.
  • In Math 3, Module 1, Topic 2, Activity 5.2, the materials provide a context involving cylindrical planters for city sidewalks and storefronts that come in a variety of sizes with specific height and radius requirements. Students generate a cubic function to model the volume $$V(x)=(\pi x^2)(2x)$$ from the base area function $$A(x) = \pi x^2$$ and the height function $$h(x) = 2x$$.

Examples where the instructional materials attend to conceptual understanding and procedural skills independently include:

  • In Math 2, Module 3, Topic 2, Activity 1.4, students develop their procedural skills in rewriting expressions involving radicals and rational exponents using the properties of exponents.
  • In Math 2, Module 4, Topic 2, Activity 3.3, Talk the Talk, students develop their conceptual understanding of systems of equations when considering the number of possible solutions for a system of equations consisting of a linear equation and a quadratic equation and two quadratic equations.

Criterion 2e - 2h

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

The instructional materials reviewed for the Carnegie Learning Math Solutions Integrated series meet the expectation for supporting the intentional development of the eight Mathematical Practices (MPs), in connection to the high school content standards. Overall, the materials integrate the use of the MPs with learning the mathematics content. Through the materials, students make sense of problems and persevere in solving, attend to precision, reason and explain, model and use tools, and make use of structure and repeated reasoning.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. MP1 and MP6 are used to enrich the mathematical content and demonstrate the full intent of these mathematical practices across the series.

The materials state that MP1 is “evident every day in every lesson” and is not explicitly identified in either the teacher or student materials. The materials provide students opportunities to explain the meaning of problems, look for entry points when problem solving, plan solution pathways, and make conjectures about the form and meaning of a solution.

Examples of where and how the materials use MP1 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 3, Topic 2, Activity 3.2, students choose an appropriate function to model data of carbon dioxide concentration in the Earth’s atmosphere over time. Students consider what information would help them make a decision as to whether a linear or exponential function is best to model this context and data.
  • In Math 2, Module 3, Topic 1, Activity 3.1, students develop a piecewise function from a scenario of pizza sales during the day. Students determine which piece should be used to determine the y-intercept of the function. 
  • In Math 3, Module 1, Topic 2, Activity 5.1, students identify what x-intercepts represent and whether their values make sense within the context of a volume formula relating height, length, and width of a planter box.

Materials use a target icon to explicitly identify MP6 in teacher and student materials. Examples of where and how the materials use MP6 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 2, Topic 4, Activity 1.2, students classify triangles on the coordinate plane as acute, right, or obtuse, and scalene, isosceles, or equilateral by calculating distances and slopes.
  • In Math 2, Module 1, Topic 1, Activity 1.1, students construct a circle, a diameter of a circle, and perpendicular bisector, and identify radii, arcs, central angles, chords, and secants using definitions of each geometric term.
  • In Math 3, Module 1, Topic 1, Activity 5.3, students use the quadratic formula to determine how long it takes for a t-shirt to land on the ground after being launched and consider whether an exact solution or approximate solution is more appropriate for the context.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. MP2 and MP3 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. The materials use a puzzle piece icon to explicitly identify MP2 and MP3 in teacher and student materials. 

Examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 2, Topic 1, Activity 5.3, students identify the quantities represented in an equation and table and determine which representation is converting Farenheit to Celsius and which representation is converting Celsius to Farenheit. Students compare the slope and y-intercept for each function within the context of the problem.
  • In Math 2, Module 3, Topic 1, Activity 5.1, students use a table to show the conversion between the U.S. dollar and the Turkish lira. Students convert Turkish lira to the U.S. dollar and consider how the quantities for this conversion relate to the original conversion as an introduction to inverses.
  • In Math 3, Module 3, Topic 3, Activity 5.2, students examine how many social media followers a quarterback and running back for a professional football team have. Students decontextualize the situation in order to think about the functions that represent the situation, and students contextualize the situation in order to interpret both equations to answer follow-up questions.

Examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 1, Topic 1, Lesson 2, Getting Started, students analyze 19 graphs, sort them into at least two different groups, and provide a rationale for how their groups were created. In Activity 2.1, students consider four different groupings students created and determine the “rule” used to create their groupings, justify why a grouping is correct based on a provided rationale, or justify why a grouping is not correct based on a provided rationale. 
  • In Math 2, Module 1, Topic 2, Activity 2.3, students analyze two students’ proof plans and determine which proof plan is correct.
  • In Math 3, Module 5, Topic 2, Activity 2.1, students explore different types of biased samples as they consider who is correct in identifying a sampling procedure as a convenience sample or a subjective sample. Students also consider a student response regarding biased samples and explain why the student’s statement is correct based on their knowledge of sampling definitions.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. MP4 and MP5 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. Materials use a wrench icon to explicitly identify MP4 and MP5 in teacher and student materials. 

Examples of where and how the materials use MP4 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 2, Topic 3, Lesson 5 Assignment, students use information about how much a baker makes when she sells decorated cookies and cupcakes, her minimum profit goal, and the maximum hours she’d like to work to create a system of linear inequalities that model the constraints. Students modify their existing constraints to account for running out of supplies.
  • In Math 2, Module 3, Topic 2, Activity 4.1, students use written descriptions of two methods of saving money. Students write a function to model each situation and add the functions. Students graph all three functions and make a connection to what students have previously learned about transformed functions.
  • In Math 3, Module 4, Topic 1, Activity 2.1, students model the height of a rider on a Ferris wheel with a periodic function. Students create a graph and a table to represent the height of a rider above the ground as a function of the number of rotations of the Ferris wheel. 

Examples of where and how MP5 is used to enrich the mathematical content and demonstrate the intentional development of the full intent of MP5 across the series include:

  • In Math 1, Module 1, Topic 3, Activity 2.2, students use technology to construct a scatter plot, determine the linear regression equation, and compute the correlation coefficient for a data set.
  • In Math 2, Module 1, Topic 1, Lesson 4, students use constructions to determine the appropriate place for building an information kiosk at a zoo. Students choose from dynamic software, a compass, or other appropriate tools. Students are not told which tool to use and are expected to choose based on availability and/or appropriateness.
  • In Math 3, Module 5, Topic 1, Activity 3.1, students use technology to find z-scores and percentiles for situations that can be modeled by normal distributions.

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 for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. MP7 and MP8 are used to enrich the mathematical content, and the materials demonstrate the full intent of these MPs across the series. The materials use a box icon to explicitly identify MP7 and MP8 in teacher and student materials.

Examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 1, Topic 2, Lesson 2, Getting Started, students consider several sequences on cut out cards, determine the unknown terms of each sequence, and describe the pattern of each sequence. Students use the patterns they observed to group the sequences and provide a rationale as to why they created each group. Students re-group the sequences as arithmetic, geometric, or neither.
  • In Math 2, Module 3, Topic 3, Activity 2.1, students use a table to calculate first and second differences for two linear equations and two quadratic equations and graph each equation. Students notice patterns to discern a relationship between the first differences for a linear function and whether the graph is increasing or decreasing as well as a relationship between the second differences for a quadratic function and whether the parabola opens up or down. 
  • In Math 3, Module 4, Topic 1, Activity 4.2, students explore patterns in the unit circle coordinates and use their knowledge of unit circle coordinates in the first quadrant and symmetry to label the coordinates in quadrants II, III, and IV.

Examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • In Math 1, Module 2, Topic 1, Activity 1.4, students express regularity in repeated reasoning to verify that constant differences in arithmetic sequences written explicitly are equivalent to the slope of the arithmetic sequence written in function notation of f(x) = ax + b.
  • In Math 2, Module 3, Topic 2, Lesson 2, Talk the Talk, students express regularity in repeated reasoning by differentiating exponential growth and decay when identifying equations that are appropriate exponential models to represent a growing population.
  • In Math 3, Module 2, Topic 3, Activity 2.1, students express regularity in repeated reasoning to sketch transformed rational functions based on the general form of transformed functions.

Gateway Three

Usability

Meets Expectations

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
8/8
+
-
Criterion Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet the expectations for being well designed and taking into account effective lesson structure and pacing. The instructional materials distinguish between problems and exercises, have exercises that are given in intentional sequences, have a variety in what students are asked to produce, and include manipulatives that are faithful representations of the mathematical objects they represent.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the underlying design of the materials distinguishes between problems and exercises. 

Each course in the series is divided into five modules and each module is divided into topics. Students engage in several lessons and activities related to that topic. The structure of each lesson is consistent throughout all courses: Students engage in Warm Ups, Getting Started tasks to activate prior knowledge, a variety of activities that allow students to develop their understanding of mathematical concepts, Talk the Talk tasks where students can reflect on and communicate their understanding of mathematical concepts they learned in the activities, and an assignment. Each assignment consists of five sections: Write, Remember, Practice, Stretch, and Review. 

MATHia provides additional personalized exercises to support student understanding of concepts. Also, Skills Practice workbooks provide exercises for students to reinforce their learning and build mastery.

Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the design of assignments is not haphazard and exercises are given in intentional sequences.

Exercises within student assignments are intentionally sequenced to build understanding and knowledge. Assignments consist of five sections:

  • Write: Students reflect on terms and concepts from the lesson,
  • Remember: Key terms and concepts from the lesson are identified for the student,
  • Practice: Students use the key concepts learned in the lesson to solve problems,
  • Stretch: Students are pushed to think beyond the skills addressed in the lesson activities to solve problems, and
  • Review: Students reinforce conceptual understanding and fluency of skills from previous lessons and topics.

Each section of the assignment reinforces student understanding of concepts from the corresponding lesson, provides practice to support student understanding of concepts from previous lessons, or assists students in making connections among concepts from several lessons.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that there is a variety in how students are asked to present the mathematics. 

Throughout lesson activities and assignments and in MATHia workspaces, students create and present mathematics in a variety of ways, such as writing and graphing equations and functions, creating and interpreting tables, analyzing and interpreting data sets, constructing geometric models, and justifying their reasoning and critiquing the reasoning of others. Examples include:

  • In Math 1, Module 1, MATHia, Introduction to Function Families Workspace, students interpret a graph and equation modeling exponential growth of a bank account earning compound interest to complete a table of outputs, identify the maximum and/or minimum of the function, explain why certain points don’t make sense given the context provided, and identify the asymptote.
  • In Math 1, Module 2, Topic 3, Activity 1.3, students write a system of linear equations to represent the amount of money Marcus and Phillip save each week, use substitution to solve the system of linear equations algebraically, and graph the system to verify their solution.
  • In Math 2, Module 3, Topic 1, Lesson 4, Talk the Talk, students write a piecewise function to model a given scenario, select one of two graphs that best represent the situation, and explain their reasoning.
  • In Math 3, Module 3, Topic 1, Activity 4.4, students compare two correct solution methods for multiplying radicals and explain the difference in their methods and critique the reasoning of an incorrect solution method for dividing radicals. Students also consider four solution methods for adding and subtracting radical terms and justify the correct methods and critique the incorrect methods.

Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Manipulatives, such as graphing technology, grid paper, patty paper, rulers, compasses, and straightedges, are embedded in activities throughout the course. Examples include:

  • In Math 1, Module 3, Topic 2, Activity 3.2, students use technology to choose the correct function (linear or exponential) and write a regression equation to model a data set.
  • In Math 2, Module 1, students use patty paper, compass, and straightedge to make geometric constructions.
  • In Math 3, Module 1, Topic 2, Activity 1.1, students draw a rectangle, disc, and isosceles triangle on an index card, cut out the shape and tape it along the center to a pencil, and then spin the pencil to rotate the shape to determine what three-dimensional solid is formed by spinning the two-dimensional shape.

MATHia workspaces incorporate animations, explore tools, classification tools, and problem solving tools (graphing tools and virtual manipulatives).

Indicator 3e

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
0/0
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series have a visual design that is not distracting or chaotic and supports students in engaging thoughtfully with the subject. The design of the materials is consistent among modules within courses across the series. The design is streamlined with large sectioned headers and highlighted worked examples with no distracting pictures or graphics. Ample white space is provided for students to make notes and perform calculations. Graphs, tables, and other figures are provided when appropriate for students to complete. Side bars are provided in the margins to highlight important information or provide helpful insights to students throughout the lessons. Examples include text boxes highlighting vocabulary and thought bubbles to help students remember a concept previously learned or think about different strategies that may be helpful.

The MATHia dashboard requires several mouse clicks to access necessary material. However, the lesson workspaces are consistent among modules within courses across the series. The design of lesson workspaces is streamlined with animations or given information on the left and exercises for students to complete on the right of the workspace. There are no distracting pictures or graphics.

Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
8/8
+
-
Criterion Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet the expectations for supporting teacher learning and understanding of the Standards. The instructional materials support teachers by: planning and providing learning experiences with quality questions; containing ample and useful notations and suggestions on how to present the content; containing full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons; and containing explanations of the grade-level mathematics in the context of the overall mathematics curriculum.

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson contains Getting Started, Activities, and Talk the Talk sections. Facilitation Notes are provided for each of these sections and contain a list of suggested questions, labeled Questions to Ask, to move student learning forward.

Also, the materials support teachers in planning and providing effective learning experiences by providing module, topic, and lesson overviews that contain guidance for ways in which students can demonstrate understanding and probing questions to be used during instruction.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for containing 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. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide detailed information regarding the instructional design of the series, lesson structure, assignment structure, problem types students will work with, thought bubbles to promote student self-reflection, mathematical habits of mind, the academic glossary, the modeling process, and assessments. Detailed information regarding content alignment within a given course complement a general overview of standards addressed within that course.

The Facilitation Notes embedded within every lesson provide ample and useful annotations and suggestions regarding an overview of the mathematical concepts addressed, standards addressed, essential ideas, pacing, what to look for from students, Questions to Ask, grouping strategies, common student misconceptions, differentiation strategies, and a summary statement of the mathematical ideas addressed.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. The introductory pages of the first volume of the Teacher’s Implementation Guide for each course provide guidance for using the MATHia software and collecting data reports generated by MATHia. The table of contents explicitly identifies certain MATHia workspaces that correlate to specific modules in the given course.

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.
2/2
+
-
Indicator Rating Details

The instructional materials for Carnegie Learning Math Solutions Integrated series  meet expectations for containing a teacher’s edition that contains full, adult-level explanation 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.

Within MyPL, teachers can view instructional videos that provide adult-level explanations and examples for teachers to enhance their own knowledge of the content. The instructional videos address textbook lessons, MATHia, mathematical content, and classroom strategies. For example, in the video, G of X (Integrated Math 1, Module 1, Topic 1, Lesson 3), teachers view suggestions for implementing the lesson. The Teacher’s Implementation Guide for each course provides detailed information regarding how mathematical content fits into the series overall, and the materials include module overviews that describe the mathematics of the module and how the content is connected to prior and future learning. MyPL also includes 33 videos addressing mathematical content that are not lesson-specific, and the advanced mathematics concepts addressed by the videos include, but are not limited to: ellipses, hyperbolas, and discontinuities and asymptotes of rational functions.

Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for containing a teacher’s edition that explains the role of the specific mathematics standards in the context of the overall series. For each course, the Module Overview provides information on how the mathematical ideas from that module connect to prior learning and future learning. The Topic Overview provides the entry point or prior experience with the mathematical ideas for students, as well as how the activities in that topic promote student expertise in the mathematical practice standards. 

Additionally, each Teacher’s Implementation Guide includes a Course Module Overview table that provides an overview of the mathematical ideas of each module, how the lessons within that module connect to students’ prior learning, and how students use the mathematical ideas from that module in later modules within the course or series. For example, in the Integrated Math 1 Module Overview for Module 3: Investigating Growth and Decay states:

  • Connections to Prior Learning: “The lessons in this module build on students’ experience with geometric sequences and common ratios. They have learned the effects of dilations and horizontal and vertical translations functions. Students have also used horizontal lines to solve linear functions graphically.”
  • Module Overview: “Students learn that certain geometric sequences can be defined as exponential functions. They transform exponential functions and distinguish between growth and decay scenarios. Students solve exponential equations algebraically using common bases or graphically using a horizontal line.” 
  • Connections to Future Learning: “Exponential functions are the first nonlinear functions that students have studied in depth. Throughout this module students compare and contrast linear and exponential functions, an important skill as students encounter more complicated function types in high school mathematics.”

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).
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series provide a list of lessons in the teacher’s edition, which cross-references the standards addressed and provides an estimated instructional time for each lesson, chapter, and unit (i.e., pacing guide). The Teacher’s Implementation Guide for each course includes a course Standards Overview chart that identifies which lessons address and/or review specific standards. A pacing guide is included at the module level, topic level, and lesson level within each course in the series. Suggested pacing for assigning MATHia workspaces is also included in the topic overviews.

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.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement. The introductory pages of the Teacher’s Implementation Guide for each course include Getting Ready that identifies ways to inform students and parents or caregivers about the mathematics program. The materials provide a Family Guide for every topic throughout the series. The Family Guide provides an overview of the mathematical ideas addressed in that topic, how the math is connected to what students already know, and how that knowledge will be used in future learning, and a few key terms students will learn. The materials state, “While we don’t expect parents to be math teachers, the Family Guides are designed to assist families as they talk to their students about what they are learning.” 

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series contain explanations of the instructional approaches of the program and identification of the research-based strategies. The introductory pages of the Teacher’s Implementation Guide for each course includes information about the Carnegie teaching and learning philosophy and design, along with the research and rationale to support the design of the materials. The instructional design is described in detail, and the Connecting Content and Practice in the Teacher’s Implementation Guide describes each feature of the materials along with suggestions for its intended use and, in some cases, connections to research.

Criterion 3m - 3q

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

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet the expectations for offering teachers resources and tools to collect ongoing data about student progress on the Standards. The instructional materials provide opportunities to collect information about students’ prior knowledge and strategies for how to utilize the information in the classroom. The materials provide opportunities for identifying and addressing common student errors and misconceptions, ongoing review and practice with feedback, and assessments with standards clearly identified. The assessments contain detailed rubrics and answer keys, and there is guidance for interpreting student performance or suggestions for follow-up.

Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing strategies for gathering information about students’ prior knowledge within and across courses. There is a pre-test for every topic in each module that addresses standards that will be taught. The post-test for the topic is similar to the pre-test, although not the same questions. The Topic Overview provides a list of Prerequisite Skills needed for the topic, which creates an indirect opportunity for teachers to gather information about students’ prior knowledge, although there is no direct guidance provided to the teacher about how to use the information. MATHia software is used as an assessment and progress monitoring tool. In every assignment in the textbook, there is a Review section. Students practice two questions from the previous lesson, two questions from the previous topic, and two questions that address the fluency standards outlined in the standards.  In the Module Overview, there is a connection to student’s prior learning. This explains to the teacher what students should know or be able to do based on previous learning.

While there are opportunities to gather information about students’ prior knowledge, the materials do not provide specific strategies about how to utilize the data in the classroom.

Indicator 3n

Materials provide support for teachers to identify and address common student errors and misconceptions.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support for teachers to identify and address common student errors and misconceptions. Materials highlight common student errors and/or misconceptions for teachers in the lesson Facilitation Notes in the Teacher’s Implementation Guide for each course. The materials also provide strategies to teachers that are mathematically sound for addressing common student errors and/or misconceptions in the lesson Facilitation Notes in the Teacher’s Implementation Guide. Examples include:

  • In Math 1, Module 2, Topic 1, Activity 3.3, Facilitation Notes, the materials identify the following misconception when students vertically dilate a linear function: “Students may look at a graph that is vertically stretched and see it as horizontally compressed. Address the fact that without more information, the graph could be described both ways; however, the function provides additional information. The A-value deals with vertical movement. Therefore, the y-values are affected by the transformation, while the x-values stay constant. Have students take notes with stretches for a graph that is vertically stretched and a graph that is vertically compressed. Horizontal transformations will be addressed with exponential and quadratic functions.”
  • In Math 3, Module 4, Topic 1, Activity 4.5, Facilitation Notes, the materials identify the following misconception when students rewrite a quadratic in vertex form to solve problems: “Students may think that because they used one method to start on the solution path that they cannot use a different method at any point. For example, when a quadratic equation is written in general form, the zeros may be easiest to identify using graphing technology. Encourage students to try different solution paths.”
  • In Math 3, Module 1, Topic 3, Activity 1.3, Facilitation Notes, the materials identify the following misconception when students investigate characteristics of even and odd functions, “Students may confuse even functions with even-degree polynomials and odd functions with odd-degree polynomials.” Materials suggest asking the following questions to help students: “What is the difference between an even function and an even-degree polynomial? What is an example of an even function that is not an even-degree polynomial? What is the difference between an odd function and an odd-degree polynomial? What is an example of an odd function that is not an odd-degree polynomial?”

Additionally, the MATHia software identifies and addresses common student errors and misconceptions. MATHia re-directs a student and provides customizable hints to address individual student difficulties.

Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide several opportunities for ongoing review and practice for students in learning both concepts and skills:

  • Warm Ups and Getting Started tasks provide review of skills from previous lessons.
  • Talk the Talk includes a few questions related to the learning students engaged in during the lesson.
  • The Assignment includes a Review of several exercises that provide ongoing practice of previous concepts and skills for students.
  • The MATHia software or, if technology is not accessible, the Skills Practice workbooks provide ongoing practice and review opportunities for students. The MATHia software continually provides feedback for correct and incorrect answers, and the Skills Practice workbooks include answers to the odd number exercises for students to check their progress. 
  • The MATHia software includes “Hints” that students can select when reviewing and practicing concepts and skills. Just-in-Time Hints automatically appear when a student makes a common error and On-Demand Hints are provided when a student asks for a hint while working on a problem. Step-by-Step Hints demonstrate how to use the tools in a lesson by providing step-by-step guidance through a sample problem.

Indicator 3p

Materials offer ongoing assessments:
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Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
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Indicator Rating Details

The instructional materials for Carnegie Learning Math Solutions Integrated series meet expectations for assessments clearly denoting which standards are being emphasized on assessments. The series provides five types of assessment for every topic: Pre-test, Post-test, End of Topic Test, Standardized Practice Test, and Performance Task. The Performance Task clearly denotes which standards are being assessed. The other assessments address the set of standards included in each topic (as outlined in the Teacher’s Implementation Guide). Materials describe which standard aligns with which problem in the Assessment Overview. Assessments do not provide standards on student-facing materials. The Edulastic Assessment Suite provides standards for each assessment, and MATHia also provides students with standards-identified content. 

Indicator 3p.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
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Indicator Rating Details

The instructional materials for Carnegie Learning Math Solutions Integrated series meet expectations for providing sufficient guidance to teachers for interpreting student performance on assessments and providing suggestions for follow-up. Materials include some guidance for teachers to interpret student performance. Answer keys are provided for all assessments. Performance Tasks include a detailed scoring rubric for teachers to use when interpreting student performance; however, no other assessment provides guidance for teachers about scoring student performance. MATHia reports provide teachers with detailed information about student performance in relation to progress on standards and suggestions on the skills that require additional support. The materials also offer teachers an APSLE (Adaptive Personalized Learning Score) report which is a predictor for year-end summative assessments. Videos within MyPL explain this report in more detail while outlining the research and models behind the report.

Indicator 3q

Materials encourage students to monitor their own progress.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series encourage students to monitor their own progress.

  • Throughout lessons in the series, materials include thought bubbles in the margin (i.e. Remember, Think About, and Ask Yourself) to encourage students to reflect on their learning. For example, in Math 3, Module 1, Topic 1, Activity 5.1, students are encouraged to think about, “What is the connection between the worked example and determining the roots from factored form, $$ y = a(x - r_1)(x - r_2)$$?”
  • In Talk the Talk, students monitor their progress on mathematical concepts and skills addressed in each lesson.
  • In Assignment, Review, students monitor their progress on mathematical concepts and skills addressed in previous lessons.
  • MATHia uses a Progress Bar to assist students in monitoring their progress. “Hints” available in MATHia encourage students to monitor their progress regarding their level of understanding of concepts and skills.

Criterion 3r - 3y

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

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet the expectations for supporting teachers in differentiating instruction for diverse learners within and across courses. The instructional materials provide strategies to help teachers sequence or scaffold lessons, strategies for meeting the needs of a range of learners, tasks with multiple entry-points, and support, accommodations, and modifications for English Language Learners and other special populations. There are opportunities for students to investigate mathematics content at greater depth, but they are intended for all students over the course of the school year with general tips for teachers to expand or deepen lessons.

Indicator 3r

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide a Content at a Glance pacing guide for each course as well as a more detailed pacing guide for lessons and activities within each module in the Teacher’s Implementation Guide. Each module overview in the Teacher’s Implementation Guide identifies how the content in that module is connected to students’ prior learning and future learning. All lessons include instructional notes and classroom strategies that provide teachers with sample questions, differentiation strategies, common student misconceptions, what to look for from students, and summary points. Each of these components provide structure for the teacher to help scaffold lessons so that the content is accessible to all learners.

Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing teachers with strategies for meeting the needs of a range of learners. MATHia, an adaptive online component of the series, differentiates learning for every learner. The level of support is individualized to each student as the program adapts to student answers. 

The Teacher’s Implementation Guide provides differentiation strategies, Questions to Ask, and common misconceptions for most lessons. Most Questions to Ask are intended for all learners, while the differentiation strategies provided are specifically intended for all students, students who struggle, or students who are ready to extend their learning. For example, in Math 1, Module 2, Topic 1, Activity 5.2, the Teacher’s Implementation Guide lists the following differentiation strategies in the Facilitation Notes: 

  • “To support students who struggle, have them make a table for the graph so that they have the same representations to compare.”
  • “To extend the activity, ask students to compare the two recycling centers using equations.”

Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. In the Instructional Design section of the Teacher’s Implementation Guides, the materials state, “Carnegie Learning recognizes the importance of connecting multiple representations of mathematical concepts. Lessons present content visually, algebraically, numerically, and verbally.”

Each Topic Overview in the Teacher’s Implementation Guide includes a section titled: “What is the entry point for students?” The Topic Overview for Math 2, Module 4, Topic 1 states, “Importantly, students have extensive experience locating solutions to equations using a graphical representation. Connecting solutions to graphs helps students to understand why quadratic equations have two solutions that are symmetric about the line of symmetry. In middle school, they used the Properties of Equality and square roots when applying the Pythagorean Theorem and when solving for a side length given the area of a figure. This serves as the launching point for solving more complicated quadratic equations.”

Performance tasks often allow for students to solve a problem using a variety of solution strategies or representations. For example, in Math 2, Module 1, Topic 2, Performance Task, students use two methods to determine the distance between Earth and a satellite in orbit. Lesson activities provide limited opportunities for students to develop their own solution path, as specific strategies are often provided for students.

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).
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing support, accommodations, and modifications for English Language Learners (ELL) and other special populations that will support their regular and active participation in learning mathematics. 

ELL Tips are regularly incorporated in lessons in the Teacher’s Implementation Guide. Most of the ELL Tips address explicitly teaching vocabulary that may pose challenges and impede problem solving for students. For example, in Math 1, Module 1, Topic 3, Activity 2.1, materials offer the tip, “Help students connect the meaning of correlation with its root word relate. Then, connect the terms positive correlation and negative correlation with the slope of the line, and no correlation with no relationship.” A limited number of ELL Tips address other strategies to help ELL students, such as honoring the use of students’ native languages, building relevant background, simplifying sentences, and modifying vocabulary.

Suggestions for supporting other special populations (i.e., struggling students and advanced students) are included in the Teacher’s Implementation Guide in the Facilitation Notes for each activity. Some of these suggestions for other special populations are general, and some are specific to the content of the lessons in which they are found.

Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.
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Indicator Rating Details

The instructional materials for Carnegie Learning Math Solutions Integrated series partially meet expectations for providing support for advanced students to investigate mathematics content at greater depth. Each assignment includes a Stretch problem. The materials state, “The Stretch section is not necessarily appropriate for all learners. Assign this to students who are ready for more advanced concepts.” The materials do not provide an assignment guide for advanced students; therefore, as designed, advanced students are completing more problems than non-advanced students in the print materials. In MATHia, advanced students complete fewer of the basic problems before they move to more advanced content, which prevents them from having to do more problems than non-advanced students.

Some of the differentiation strategies listed in the Teacher’s Implementation Guide are intended to extend the activity, yet they can benefit all students. For example, in Math 2, Module 3, Topic 2, Activity 2.1, the Teacher’s Implementation Guide suggests, “To extend the activity, ask students to create posters for classroom display that highlight an increasing linear function in general form used to model simple interest and an increasing exponential function in general form used to model compound interest. Make sure they include the equations for each function and define all variables.”

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series provide a balanced portrayal of various demographic and personal characteristics. Problem scenarios vary across urban, suburban, and rural environments and should be familiar to high school students regardless of backgrounds. The names of people in word problems are representative of a variety of cultural backgrounds. 

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series provide opportunities for teachers to use a variety of grouping strategies. Carnegie Learning believes in “Learning Together” and “Learning Individually.” As stated in the Teacher’s Implementation Guides, students learn together “with our consumable textbooks, students work in groups, not only to develop skills, but to learn how to collaborate, create, communicate and problem solve” and learn individually through MATHia and/or the Skills Practice workbook. 

Facilitation notes in the Teacher’s Implementation Guide suggest grouping strategies in lessons throughout the series. For example, in Math 3, Module 1, Topic 1, Activity 1.1, materials suggest, “Have students work with a partner or in a group to complete Questions 1 and 2. Share responses as a class.”

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series partially encourage teachers to draw upon home language and culture to facilitate learning.

  • There is limited evidence of teachers drawing upon home language and culture to facilitate learning.
  • There is a Family Guide corresponding to each Topic that explains the key mathematical concepts students will be learning and provides tips to support student learning, however, it does not draw upon the home language and culture of students.
  • Materials are available in Spanish.

Criterion 3z - 3ad

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

The instructional materials for the Carnegie Learning Math Solutions Integrated series integrate technology in ways that engage students in the Mathematical Practices. The digital materials are web-based and compatible with multiple internet browsers, and they include opportunities to assess students' mathematical understandings and knowledge of procedural skills. The instructional materials include opportunities for teachers to personalize learning for all students, and the materials offer opportunities for customized, local use. However, the instructional materials do not include opportunities for teachers and/or students to collaborate with each other.

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.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series integrate technology in ways that engage students in the Mathematical Practices. MATHia includes the following interaction tools:

  • Explore Tools to investigate different mathematical concepts, search for patterns, and look for structure.
  • Animations to watch, pause, and re-watch demonstrations of various mathematical concepts.
  • Classification Tools to categorize answers based on similarities.
  • Problem Solving Tools to provide students with individualized and self-paced instruction that adapts to their needs.
  • Worked Examples to allow students to identify their own misconceptions and make sense of various mathematical concepts.

MATHia engages students in the mathematical practices by providing helpful hints, a glossary, and a progress bar in which students can track their learning.

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.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series are web-based and compatible with multiple internet browsers.  Digital requirements for the use of MATHia include (as of the 2019-2020 school year):

  • Windows computers with operating systems Windows 7 and 10,
  • Apple Computers with operating systems MAC OS X 10.13 or higher,
  • Apple iPads with iOS 11 or higher,
  • Android Tablets with Android 9 and above, or
  • Chromebooks with ChromeOS 74 or higher.
  • Compatible with Chrome, Safari, and Firefox.

MATHia is not recommended for use on phones and other small devices.

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series formatively assess students’ mathematical understanding and knowledge of procedural skills using MATHia’s Adaptive Personalized Learning Reports. MATHia can generate all of the following reports:

  • APLSE (The Adaptive Personalized Learning Score) report is a predictive report that displays class and student progress over time.
  • Session report gives a day-to-day view of work being completed by students.
  • Standards report provides an overview of how well students are mastering, or have mastered, specific standards.
  • Student Overall report provides information about the class and student progress at the module, unit, and workspace levels.

These reports provide information used to assess student learning and adjust instruction. MATHia does not summatively assess student understanding. 

Indicator 3ac

Materials can be easily customized for individual learners.
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Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series include multiple opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. The MATHia software is customizable for individual learners. Teachers may assign workspaces to one, many, or all students and workspaces can be assigned in any order. Students may progress through an assigned workspace at their own pace.

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.
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series can be customized for local use. Using the MATHia software, teachers choose which workspaces to assign to individual students or an entire class. Workspaces can be assigned in any order; however, the student must complete the workspace before beginning work in another workspace. While teachers may select workspaces to assign to students in the MATHia software, teachers cannot change the context or wording of problems.

Indicator 3ad

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.).
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Indicator Rating Details

The instructional materials for the Carnegie Learning Math Solutions Integrated series provide opportunities for teachers to communicate with a community of teachers. The materials state, “You’re part of a collective and have access to special content, events, meetups, book clubs, and more.” Teachers can access the online community through www.longlivemath.com. However, the materials do not allow opportunities for students to collaborate with each other online or through technology-based programs.

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Additional Publication Details

Report Published Date: 10/02/2019

Report Edition: 2018

Title ISBN Edition Publisher Year
Print HSMS SE & MATHia Bundle Integrated Math 1 978-1-60972-923-3 Carnegie Learning 2018
Print HSMS SE & MATHia Bundle Integrated Math 2 978-1-60972-924-0 Carnegie Learning 2018
Print HSMS SE & MATHia Bundle Integrated Math 3 978-1-60972-925-7 Carnegie Learning 2018

About Publishers Responses

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.

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.

Advancing Through Gateways

  • 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.

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