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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 Integrated 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 Integrated 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 Carnegie Learning High School Math Solution 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, Lesson 4, Activity 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 building on the given information from Lesson 4, Getting Started . 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, Activity 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, Topic 1, 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, Lesson 2, Activity 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, Lesson 2, Activity 4, students use linear combinations to solve a system of two equations in two variables within the context of selling two types of bracelets at a school store. Students are provided two different solution pathways as first steps for solving the system of equations using elimination, determine which solution pathway is correct, and justify their reasoning. Students then solve the system of equations and check their solution algebraically to confirm that linear combinations produce a correct solution for that particular system of equations. There was no evidence found where the materials or students prove that this method is true for other systems of equations.
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 Integrated series meet 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.
The following examples use the full intent of the modeling process:
In Math 1, Module 2, Topic 3, Lesson 5, Activity 1, students are given two plans a bicycle company is considering to make a low price ultra-light bicycle. Students have been “hired” to analyze the costs for each proposed bicycle prototype and determine which plan the company should follow. Students create a system of linear equations representing the scenario and have choice in terms of their solution method. Students are able to conclude either plan is the “best” plan based on the number of bicycles being produced and their mathematical findings (A-CED.3).
In Math 1, Module 3, Topic 2, Lesson 4, Activity 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 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 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 (G-GMD.3).
In Math 3, 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 Math 3, 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 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 2, Module 3, Topic 2, Lesson 1 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, Lesson 2, Activity 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, Lesson 2, Activity 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, Lesson 2, Activity 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, Lesson 1, Activity 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, Lesson 3, Activity 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, Lesson 1, Activity 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, Lesson 3, Activities 2 and 3; Math 2, Module 1, Topic 3, Lesson 3, Activity 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
Materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for the Carnegie Learning High School Math Solution 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, 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 Math 2, Module 4, Topic 1, Lesson 5, Activity 1, the materials derive the quadratic formula by providing intensive scaffolds where students fill in the blank steps for 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, Lesson 1, Activity 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, Lesson 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 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, Lesson 1, Activity 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, Lesson 1, Activity 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, Lesson 6, Activity 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, Lesson 2, Activities 2, 3, and 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, Lesson 3, Talk the Talk, 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, Lesson 5, Activities 1 and 2, students expand upon their knowledge of square roots and cube roots to solve rational equations.
In Math 3, Module 4, Topic 1, Lesson 3, Activity 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, Lesson 2, Activity 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, Lesson 5, Activity 1, students calculate the 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, Lesson 3, Talk the Talk 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, Lesson 4, Activity 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
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 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, Lesson 2, Activity 1, students analyze patterns in sequences and then formally identify sequences as arithmetic or geometric. In Module 1, Topic 2, Lesson 2, Activity 2, students match sequences to their appropriate graphs and verify that all sequences are functions. In Module 2, Topic 1, Lesson 1, Activity 1, students use their knowledge of arithmetic sequences to write a linear function in the form f(x)=ax+bf(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, Lesson 4, Activity 4, students complete the square to determine the roots of a quadratic equation. In Lesson 4, Activity 5, students rewrite a quadratic equation to identify the axis of symmetry and the vertex. In Lesson 5, Activity 1, students derive the quadratic formula. In Topic 3, Lesson 1, Activity 2, students write the general equation of a circle. (A-SSE.3a,b)
In Math 3, Module 1, Topic 2, Lesson 4, Activity 1, students build a cubic function from a quadratic and linear function. In Module 1, Topic 2, Lesson 6, Activity 2, students decompose a cubic function into three linear functions. In Module 1, Topic 3, students graph and analyze key characteristics of polynomial 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, Lesson 1, Activity 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, Lesson 2, Activity 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, 1 Lesson 3, which is later used in Math 3, Module 4, Topic 1, Lesson 1, Activity 1 to derive the formula A=ab sin(C) for the area of a triangle. 2
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
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 Integrated 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 Math 1, Module 3, Getting Ready, the materials state, “You will extend your work with geometric sequences and common ratios to develop exponential functions. You will explore using common bases to solve exponential equations. You will transform exponential functions. You will distinguish between exponential growth and decay and solve real-world problems modeled by exponential functions. The lesson in this module build on your prior experiences with exponents, transformations, and the Properties of Powers.”
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 Math 2, Module 2, Topic 2, Lesson 4, an Lesson Opener Review has students calculate the decimal equivalents of radicals. “Use a calculator to compute each. Round your answers to the nearest hundredth. 1. \frac{\sqrt{3}}{3} 2. 2\sqrt{3} 3. \frac{\sqrt{2}}{2} 4. 3\sqrt{2}”. The Teacher Implementation Guide states, “In this Review. Students calculate the decimal equivalents of radicals. They will use this skill in ACTIVITY 1 Connecting Slope and Tangent.”
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 Math 3, Module 4, Topic 2, Lesson 3, the materials state, “You have explored how the values of the transformed function form affect the shape of the graph of a periodic function. How can you use what you know to build a trigonometric function to model circular motion in real-world problems?” 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 Math 1, Module 3, Topic 1, Mixed Practice, Spaced Review, Question 1 and Question 2 are aligned to 8.EE.8 and have students write and solve a system of linear equations about provided situations.
In Math 2, Module 5, Topic 2, Mixed Practice, Spaced Review, Question 1 is aligned to 7.SP.8b and has students sketch a tree diagram to represent the sample space for a provided situation. Question 2 is aligned to 7.SP.8 and has students identify the sample space and determine the probability for a provided situation. Question 7 is aligned to 7.SP.8b and has students identify the sample space and determine the size of the sample space using the Counting Principle for a provided situation.
In Math 3, 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 Math 2, Module 1, Topic 1, Topic Overview, the materials state, “Throughout elementary and middle school, students have informally investigated many of the relationships explored in this topic. In grade 8, they have used informal arguments to establish facts about angle pairs created when parallel lines are cut by a transversal.” In this topic, students make constructions to reason about relationships when a transversal cuts parallel lines.
In Math 3, Module 1, Topic 2, Topic Overview, the materials state, “In middle school, students explored the cross sections of three-dimensional figures. They sliced rectangular prisms and pyramids and identified the polygon represented by the cross-section.” In this topic, students will move between two-dimensional and three dimensional figures, as they consider the connection between degree-1, degree-2, and higher-order polynomials.
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 Math 1, Module 2, Topic 1, Family Guide, the materials state, “ Where have we been? 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.” “Where are we going? Students should understand the key characteristics of a linear function represented in situations, tables, equations, and graphs. Solving equations using horizontal lines on the graph lays the foundation for solving systems of linear equations as well as the more complicated nonlinear equations.” .
In Math 2, Module 4, Topic 3, Family Guide, the materials state, “ Where have we been? Students have used the Pythagorean Theorem to solve for distances on the coordinate plane, to derive the Distance Formula, and to verify properties of triangles and quadrilaterals on a coordinate plane...” “Where are we going? Conic sections such as circles and parabolas will be useful in studying three dimensional geometry. Conic sections model important physical processes in nature…”
In Math 3, Module 2, Topic 3, Family Guide, the materials state, “Where have we been? Students have been working with rational numbers since elementary school. They have extensive knowledge of function behaviors and characteristics to apply to the analysis of rational functions.” “Where are we going? Rational functions are used heavily in medical and econometric modeling applications for analysis and prediction. Rational functions also have applications in image resolution and acoustics.”
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.