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Focus & Coherence
Gateway 1 - Meets Expectations | 88% |
|---|---|
Criterion 1.1: Focus & Coherence | 16 / 18 |
Criterion 1.1: Focus & Coherence
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).
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.*
Indicator 1a.i
The 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 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.
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.
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.
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.
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.
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.
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.
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.