High School - Gateway 1
Back to High School Overview
Note on review tool versions
See the series overview page to confirm the review tool version used to create this report.
- Our current review tool version is 2.0. Learn more
- Reports conducted using earlier review tools (v1.0 and v1.5) contain valuable insights but may not fully align with our current instructional priorities. Read our guide to using earlier reports and review tools
Loading navigation...
Focus & Coherence
Gateway 1 - Meets Expectations | 94% |
|---|---|
Criterion 1.1: Focus and Coherence | 17 / 18 |
The materials reviewed for Carnegie Learning High School Math Solution Traditional series meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content for all students; attending to the full intent of the modeling process; spending the majority of time on content widely applicable as prerequisites; allowing students to fully learn each standard; engaging students in mathematics at a level of sophistication appropriate to high school; and making meaningful connections in a single course and throughout the series. The materials partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.
Criterion 1.1: Focus and Coherence
Materials are coherent and consistent with “the high school standards that specify the mathematics which all students should study in order to be college and career ready” (p. 57 CCSSM).
The materials reviewed for Carnegie Learning High School Math Solution Traditional series meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content for all students; attending to the full intent of the modeling process; spending the majority of time on content widely applicable as prerequisites; allowing students to fully learn each standard; engaging students in mathematics at a level of sophistication appropriate to high school; and making meaningful connections in a single course and throughout the series. The materials partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.
Indicator 1a
Materials focus on the high school standards.
Indicator 1a.i
Materials attend to the full intent of the mathematical content contained in the high school standards for all students.
The instructional materials reviewed for the Carnegie Learning Math Solutions Traditional 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 addressed by the courses of the series:
N-RN.3: In Algebra I, Module 3, Topic 1, Lesson 2, Activity 5, students explain how the properties of rational exponents extend from the properties of integer exponents. Students consider calculations between numbers and determine to which number set the answer belongs. For example, students determine if the product of a nonzero rational number and an irrational number is sometimes, always, or never equal to a rational number.
A-REI.1: In Algebra I, Module 2, Topics 1 and 2, students use properties of equality to solve equations and inequalities. Students check and justify their solutions.
F-IF.7c: In Algebra II, Module 1, Topic 2, students identify zeros, predict the shape of a function by sketching the graph and labeling the zeros, and check their solutions using a graphing calculator. Students use technology to graph, identify domain and range, determine intervals of increasing and decreasing, and the x- and y-intercepts. In the same module, Topic 3, Lesson 3, Activity 2, students encounter the definition of end behavior and complete a table to describe end behavior.
G-CO.3: In Geometry, Module 1, Topic 2 Lessons 1-5, students describe symmetry in a rectangle and reflectional symmetry in a square. Students also determine if shapes have rotational or reflectional symmetry and are asked to “describe the reflections and rotations that can carry each figure onto itself.”
S-CP.3: In Geometry, Module 5, Topic 2, students use conditional probability to solve problems and identify independent or dependent events. Also in the topic, students use an example of conditional probability and determine if the probability of A given B is the same as the probability of A.
The following non-plus standards are not fully addressed:
A-REI.5: There was no evidence found where the materials or students prove, or are shown how to prove, that given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Indicator 1a.ii
Materials attend to the full intent of the modeling process when applied to the modeling standards.
The materials reviewed for the Carnegie Learning High School Math Solution Traditional series meets expectations for attending to the full intent of the modeling process when applied to the modeling standards. Materials intentionally develop the full intent of the modeling process throughout the series leading to culminating experiences that address all, or nearly all, of the modeling standards.
Examples where the full intent of the modeling process is used to address modeling standards across the courses of the series include:
In Algebra I, Module 1, Topic 2, Lesson 4, Getting Started, students play a game involving moving an entire stack of discs or coins from a starting circle to any of the other circles while only moving one disc/coin at a time, and not putting a larger disc on top of a smaller disc. In Activity 1, students record their results from the game in a table, identify patterns in their results, and use mathematical notation to represent their pattern. Students make predictions about how accurate the pattern holds if additional discs/coins are added and then play the game to test their prediction. If student predictions are incorrect, they are encouraged to revisit their mathematical model and make adjustments or to start all over with a new model. Students write a conclusion stating the relationship between the number of discs and the minimum number of moves to complete the game (F-BF.2).
In Algebra I, Module 3, Topic 2, Lesson 4, students use data relating a driver’s Blood Alcohol Content (BAC) and the probability of a driver causing an accident to create a model predicting 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 Lesson 4, Activity 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) (N-Q.2 and S-ID.6a).
In Geometry, Module 3, Topic 2, Assessments, Performance Task, Trigonometry: “It’s a Bird! It’s a Plane! It’s … a Drone?”students are given information about drones flown by Jeremy and Leslie. Students determine which drone is higher, the difference between the heights, and the distance between Jeremy’s and Leslie’s locations on the ground. Students consider if Jeremy flies his drone the same height as Leslie’s drone, and they explain why the angle of elevation from Jeremy’s location is different from the angle of elevation between Leslie’s location on the ground and her drone. Students calculate the angle of elevation from Jeremy’s location on the ground to his drone to validate their reasoning. According to the rubric, students provide a labeled drawing, including all given measurements and calculated measurements. Students formulate their own equation in order to compute whose drone is higher and the difference between the heights of the drone. Students compute the distance between Jeremy and Leslie with supporting work. Students provide an explanation as well as validation to support their work (G-SRT.1).
In Algebra II, Module 3, Topic 3, Performance Task, Exponential and Logarithmic Equations: “Bug Off!”students are given information about a particular insect. Students determine how many insects they could be at the end of a certain year based on the insect current population, year discovered and how much the insects continually increase by. Additionally, students determine when the number of insects would reach one million, and analyze another group of scientists' work to calculate what monthly rate they are using to predict their number of insects after a certain period of time. Students are introduced to a new group of insects with its own set of characteristics and must determine after how many months would the two populations of insects be equal. Students provide an explanation as well as validation to support their work (A.REI.11, F.BF.5,F.LE.4).
In Algebra II, Module 5, Topic 2, Lesson 1, Activity 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 Lesson 2, Talk the Talk, students select a sampling method and conduct their survey. In Lesson 3, Activity 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 Lesson 4, Activity 4, students apply their calculations from Lesson 3, Activity 4 and use statistical significance to make inferences about the population based on their collected data. In Lesson 5, Activity 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 (S-IC.1-6).
Indicator 1b
Materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
Indicator 1b.i
Materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.
The instructional materials reviewed for the Carnegie Learning High School Math Solution Traditional series, when used as designed, meet expectations for allowing students to spend the majority of their time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers. The materials do not include distracting content that would keep students from engaging with the standards identified as widely applicable prerequisites (WAPs).
Examples of how the materials allow students to spend the majority of their time on the WAPs include, but are not limited to:
In Algebra I, Module 3, Topic 2, Lesson 2, students rewrite radical expressions and rational expressions using the properties of exponents. In Geometry, Module 2, Topic 2, Lesson 4, students use and rewrite rational expressions arising from trigonometric ratios in special right triangles. In Algebra II, Module 3, Topic 1, Lesson 4, students write radical expressions with rational exponents, rewrite the radicands of variable expressions, and compute the sums/differences of irrational numbers in algebraic expressions. (N-RN)
In Algebra 1, Module 5, Topic 1, Lesson 2, Activity 4, students factor quadratic expressions and graph them in order to find the zeros (A SSE.3a). Also, in Algebra 1, Module 1, Topic 2, Lesson 4, students complete the square to transform quadratics into vertex form and find zeros in order to graph quadratics (A-SSE.3b).
In Algebra I in the MATHia software, students explore constant change, evaluate linear functions, complete charts, identify the input value, compute the equation by substituting, and determine the output value. In Algebra II, Module 1, Topic 3, students explore the characteristics of polynomial functions. In Algebra II, Module 3, Topic 4, students write equations of exponential functions from patterns. In Algebra II, Module 4, students explore trigonometric functions. (F-IF)
In Geometry, Module 3, Topic 1, students use dilations to create similar figures and establish criteria for determining similar triangles. Students use similarity to establish proportionality theorems and use similar triangles to solve problems (G-SRT.A, G-SRT.B). In Geometry, Module 3, Topic 2, students use similarity to determine constant ratios in right triangles and define trigonometric ratios (G-SRT.C).
The Statistics WAPs are addressed in Algebra I and Algebra II, and students have multiple opportunities to engage with the WAPs from this category.
Indicator 1b.ii
Materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for the Carnegie Learning High School Math Solution Traditional series, when used as designed, meet expectations for letting students fully learn each non-plus standard. However, the instructional materials for the series, when used as designed, do not enable students to fully learn a few of the non-plus standards.
Examples of the non-plus standards that would not be fully learned by students when using the materials as intended include:
A-SSE.4: In Algebra 2, Module 3, Topic 4, Lesson 1, Activity 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 Algebra 1, Module 5, Topic 2, Lesson 5, students complete the square in order to solve quadratic equations. The materials derive the quadratic formula by completing the square, but 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 Algebra 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 Algebra 1, Module 5, Topic 1, Lesson 1, Activity 3, students find the intersection of constant and quadratic equations and explain why the x- and y-coordinates of the points where the graphs intersect are solutions. In Algebra 1, Module 5, Topic 3, Lesson 2, 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 Geometry, Module 4, Topic 1, 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 at the beginning of Lesson 2, Activity 1.
Indicator 1c
Materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed for the Carnegie Learning High School Math Solution Traditional series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from grades 6-8.
Examples of applying key takeaways include, but are not limited to:
In Algebra I, Module 1, Topic 1, students interpret key features of graphs in the context of a problem (8.F.4). In Lesson 1, Activity 1, students read scenarios, determine the graph which represents the scenario, and identify independent and dependent quantities. In Lesson 1, Activity 2, students compare and contrast the graphs they organized in the first sections.
In Geometry, Module 3, Topic 1, students develop similarity standards (8.G.4). In Lesson 1, Activity 1, students dilate figures to create similar figures. In Activity Lesson 1, Activity 2, students establish similarity criteria. In Lesson 5, Activity 1, students apply knowledge of similar triangles in order to solve mathematical problems.
In Algebra I, Module 5, Topic 2, Lesson 5, students solve problems and apply knowledge of real numbers to determine to which set the answers belong. In Algebra II, Module 1, Topic 1, Lesson 6, students apply knowledge of real numbers to determine if solutions are a part of the real number system. These activities apply key takeaways from 6-8.NS involving integers, rational numbers, and irrational numbers.
Examples of age appropriate contexts throughout the series include, but are not limited to:
In Algebra I, Module 1, Topic 3, Lesson 3, Activity 1, students construct a scatter plot that relates speed and braking distance.
In Geometry, Module 3, Topic 2, Lesson 2, Talk the Talk task, students relate road grades of mountainous areas to the angle of elevation.
In Algebra II, Module 1, Topic 1, Lesson 1, Activity 2, students use a scenario regarding homecoming elections and how the results have been shared. Also in this lesson, students explore the pattern of a rumor spreading.
In Algebra II, Module 1, Topic 1, Lesson 5, Activity 3, students apply a quadratic model to launching a free t-shirt to fans at a baseball game.
The instructional materials use various types of numbers throughout the series in expected places where numbers are naturally varied, such as trigonometry, regression models, and problems involving radicals and/or rational exponents. Examples of the materials using various types of numbers include, but are not limited to:
In Algebra I, Module 5, Topic 2, Lesson 5, Activity 4, students evaluate the discriminant of quadratic equations and the calculations include real, rational, irrational, and imaginary numbers.
In Geometry, Module 4, Topic 2, Lesson 3, Activity 2, students determine the intersection points for each system of equations involving a circle, including rational and irrational coordinates.
In Algebra II, Module 4, Topic 2, students use integers, rational numbers, and irrational numbers in trigonometric functions and relationships.
Indicator 1d
Materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
The instructional materials reviewed for the Carnegie Learning High School Math Solution Traditional series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials foster coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards.
Examples of connections made across courses include, but are not limited to:
In Algebra I, Module 3, Topic 1, and Algebra II, Module 3, Topic 1, students use rational exponents and their definitions in order to solve problems (N-RN.1).
In Algebra I, Module 5, Topics 1and 2, and Algebra II, Module 2, Topic 1, factoring, completing the square, and finding equivalent forms are connected. Students write equivalent forms and make connections between solutions, graphs, and other key details using the equivalent forms (i.e., connections between factors and zeros and completing the square to reveal the vertex or as an alternate way of solving an equation) (A-SSE.3).
In Algebra I, Topic 1 of Modules 1, 2, 3, and 5, and Algebra II, Module 2, Topic 1, Module 3, Topics 1-3, and Module 4, students use functions to model different situations. Although often confined to a specific type of function, students complete a process to develop understanding of why a certain function type is used in a context (F-IF.4).
In Algebra II, Module 1, Topic 3, and Module 2, Topic 3, Lesson 3, students transform rational and polynomial functions and sketch a graph involving a transformed function. Students apply knowledge of geometric transformations from Geometry Module 1 to transform a graph based on an equation (G-CO).
In Geometry, Module 4, Topic 2, Lesson 2, students use completing the square as a method for writing the equation of a circle. Students also use completing the square as a way to calculate minimum and maximum values in Algebra I, Module 5, Topics 6 and 7, and in Geometry, they determine the equation of a circle in standard form using the same method (A-SSE.B).
The following examples are instances where meaningful connections are made within courses:
In Algebra I, average rate of change is addressed through linear functions in Module 2 and students explore average rate of change with exponentials in Module 3, including connections to a constant ratio for geometric sequences and the use of MATHia Software. Average rate of change is explored for quadratic functions in Module 5 (F-IF.6).
Transformations are used throughout the Geometry course. In Module 1, Topic 1, students are introduced to the idea of transformations where definitions for both transformations and rigid motion are given. In Topic 3, students explore the different rigid motions using simple geometric transformation machines, similar to function machines. In Module 2, Topic 1, Lesson 2, students use transformations to prove triangle congruence, and in Lesson 3 students use triangle congruence to solve problems (G-CO).
In Algebra II, students connect patterns to functions, tables, and graphs. In Module 1, students match scenarios, various function forms, tables, and graphs with linear, exponential, and quadratic functions. In Module 2, students factor, use long division, sketch graphs, and analyze tables. Students also write equations to solve problems from scenarios (A-SSE).
Indicator 1e
Materials explicitly identify and build on knowledge from Grades 6-8 to the high school standards.
The materials reviewed for the Carnegie Learning High School Math Solution Traditional series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Although the materials explicitly identify the Grade 6-8 readiness standards being built upon in some lessons in The Coherence Maps, the connections between those standards from Grades 6-8 and high school standards are not clear.
The following are examples where the materials do not explicitly identify Grades 6-8 standards:
The Getting Ready states how the lessons in the module build upon students prior experiences and what they will be learning. However, the materials do not explicitly identify Grades 6-8 standards. In Algebra I, Module 1, Getting Ready, the materials state, “You will investigate a variety of function families, recognizing their key characteristics. You will build on your work with numeric patterns to explore sequences as functions. You will expand your knowledge of lines of best fit to determine whether it is appropriate to use a linear model and differentiation between correlation and causation. The lessons in this module build on your prior experiences with functions, numeric patterns, and writing linear equations.”
The “In This Review” are questions that present a formative assessment opportunity, building from the Getting Ready, that reactivates students’ prior knowledge. Although the Lesson “In This Review” questions draw from students' experiences in Grades 6-8, the questions do not explicitly identify content from Grades 6-8. In Geometry, Module 3, Topic 2, Lesson 4, an Lesson Opener Review has students answer questions based on a triangle NPS. “Consider \triangle NPS. 1. Calculate the value of \sin P. 2. Calculate m\angle P. 3. Calculate m\angle N.” The Teacher Implementation Guide states, “In this Review. Students practice with the sine ratio. They will use this skill in ACTIVITY 1 The Cosine Ratio.”
The Teacher Implementation Guide states, that a Teacher can “Tap into your students’ prior learning by reading the narrative statement”. Narrative Statements can be found at the beginning of each lesson. In Algebra II, Module 4, Topic 1, Lesson 1, the materials state, “You have learned about many different types of functions. What functions can you define using points on the circle as the domain?” The Narrative Statement does not explicitly identify content from Grades 6-8.
The Teacher Implementation Guide includes a Standards Overview located in the Front Matter for each course. The Standards Overview shows which standards are covered in the lesson and the standards covered in spaced practice. Some of the spaced practice standards identified are from Grades 6-8, however it is not mentioned in the materials how these standards support the progression of the high school standards. Some spaced practice standards are addressed in the spaced review part of the mixed practice section. The materials provide an “Aligned Standards” box to identify which standard(s) are aligned to each question. Examples include but are not limited to:
In Algebra I, Module 1, Topic 3, Mixed Practice, Spaced Review, Question 1 is aligned to 8.EE.6 and has students calculate the slope of a line represented in a table. Question 2 is aligned to 8.EE.6 and has students determine whether the slope of an equation is positive, negative, zero, or undefined. Question 3 is aligned to 8.EE.7 and has students solve two-sided equations. Question 4 is aligned to 8.SP.1 and has students create a scatter plot given a data set, draw a line of best fit, and then write the equation for the line.
In Geometry, Module 1, Topic 2, Mixed Practice, Spaced Review, Question 3 is aligned to 8.F.1 and has students determine whether a mapping represents a function. Question 6 is aligned to 8.G.7 and has students determine the value of a diagonal of a rectangular prism.
In Algebra II, Module 2, Topic 3, Mixed Practice, Spaced Review, Question 2 is aligned to 8.G.7 and has students verify that a triangular piece of metal with the given side lengths is a right triangle, and if they can use Euclid’s formula to generate the given side lengths.
The Topic Overview includes, “What is the entry point for students?” in the “Connection to Prior Learning” section, although the connection to prior learning is explicitly stated, how it connects to the current topic is not always stated. Additionally, any Grades 6-8 content referenced in this section of the topic overview is not explicitly connected to specific standards. Examples from the Topic Overview include but are not limited to:
In Algebra I, Module 2, Topic 2, Topic Overview, the materials state, “In grade 7, students solved two-step inequalities and graphed the solutions on a number line.” Students build from this knowledge in this topic when they solve more complex inequalities in one variable and compound inequalities.
In Geometry, Module 1, Topic 3, Topic Overview, the materials state, “In middle school, students constructed triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, or no triangle. Through hands-on exploration, they developed an intuition regarding the minimum criteria for determining whether triangles are congruent.” Students build from this knowledge in this topic when they use triangle congruence theorems to prove congruent triangles.
The materials include a family guide with “Where have we been? Where are we going?” for each topic. These sections identify connections between middle school content and courses and lessons in the series, however, any Grade 6-8 content referenced in this section of the family guide is not explicitly connected to specific standards. Examples from the Family Guide include but are not limited to:
In Algebra I, Module 2 ,Topic 3, Family Guide, the materials state, “Where have we been? Students know that every point on the graph of an equation represents a value that makes the equation true…Students have written and solved systems of linear equations.” “Where are we going? Knowing how to solve systems of linear equations prepares students to solve systems that include nonlinear equations…”
In Geometry, Module 3, Topic 1, Family Guide, the materials state, “Where have we been? Students have developed an understanding of proportional reasoning by exploring multiplicative relationships and having used scale factors to solve problems. They have learned that two figures are similar; they can map one onto the other by a sequence of transformations.” “Where are we going? Understanding similarity further develops proportional reasoning. It provides the opportunity for students to connect spatial and numeric reasoning and lays the groundwork for understanding trigonometric ratios, which students will explore in the next topic.”
In Algebra II, Module 2, Topic 1, Family Guide, the materials state, “Where have we been? Students have factored expressions since elementary school. They have factored degree-2 polynomial equations in order to isolate key characteristics of the functions represented by those equations.” “Where are we going? … Students will use what they learned in this topic to help them build real-world polynomial models in the next topic.”
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.