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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 Agile Mind Integrated series meet the expectation for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. Overall, the instructional materials attend to the full intent of the non-plus standards and allow students to fully learn each non-plus standard, but they do not attend to the full intent of the modeling process when applied to the modeling standards. The materials have students engage in mathematics at a level of sophistication appropriate to high school and foster coherence through meaningful connections in a single course and throughout the series, but the materials do not explicitly identify knowledge from Grades 6-8. The instructional materials spend a majority of time on the widely applicable prerequisites from the CCSSM.
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 Agile Mind Integrated series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Overall, the materials address all aspects of the non-plus standards.
Examples of standards that are addressed include:
- A-SSE.2: Students use structure to rewrite equations. The materials emphasize factoring with area models in Mathematics II, Topic 3, Student Activity Sheet 4. In Mathematics III, Topic 6, Student Activity Sheet 2, students identify the structure of quadratic equations through several guided problems. In Mathematics III, Topic 6, Student Activity Sheet 4, students factor with area models and grouping, and students choose their strategy for factoring to solve equations.
- F-TF.1: In Mathematics III, Topic 16, Exploring, students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Students define radians and use radians in multiple real-world application problems.
- G-CO.10: In the materials, students encounter several theorems about triangles in a variety of formats. In Mathematics II, Topic 10, Student Activity Sheet 4, students examine a flowchart proof of the Triangle Sum Theorem, a two-column proof that two angles in a right triangle are complementary, and a paragraph proof of the Exterior Angle Theorem.
- G-CO.12: Students explore geometric constructions with a variety of tools and methods. The materials demonstrate straightedge and compass constructions in Mathematics I, Topic 18, and patty paper constructions in Mathematics I, Topic 15. In Mathematics I, Topic 18, Student Activity Sheet 1, students consider the differences between a drawing and a construction.
- N-RN.2: Students rewrite expressions involving radicals and rational exponents using the properties of exponents in Mathematics II, Topic 16, Student Activity Sheets 1-3.
The following standard is partially addressed:
- G-C.5: The formula for the arc length of a sector is stated in Mathematics II, Topic 24, Exploring Areas of Sectors and Segments, but never derived.
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 Agile Mind Integrated series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials include aspects of the modeling process in isolation or combinations, however, opportunities to engage in the full modeling process are absent across the courses of the series.
In topics indicated as containing modeling there are many real world problems, however students are given much of the information and the lessons are scaffolded with prompts that lead the student to the solution rather than providing the opportunity to reason through the problem and to apply their own approach, assumptions, and way to solve the problem that allows one to fully engage in the modeling process.
The following examples allow students to engage in aspects, but not all, of the modeling process:
- In Mathematics I, Topic 2 Assessment, students determine the amount of fencing needed. The assessment provides multiple prompts which guide students through the problem. Students do not determine important information, identify assumptions, or make predictions.
- In Mathematics I, Topic 3, Exploring Modeling with Functions, students investigate the cost of purchasing roses through different flower shops for a fundraiser. The problem includes a list of factors for students to consider, but students do not make their own assumptions as part of the modeling process because of the provided list of factors. The materials provide students with a specific method for solving the problem, and an equation is given to the students rather than allowing students to make sense of the relationship between the variables.
- In Mathematics II, Topic 4, Exploring, students find the dimensions of a garden, but students do not formulate a model. The variables are explicitly stated in the activity and the problem has a single solution method with a single correct answer. Students do not interpret or validate their results.
- In Mathematics II, Topic 6, Student Activity Sheet 3, problems 2-6, students explore a real-world scenario through a series of specific questions that leads to a solution. Students do not make assumptions, interpret their model, or determine the reasonableness of their results.
- In Mathematics II, Topic 7, Assessment, students write an equation to represent the shape of the Gateway Arch in St. Louis. Students do not determine their own way to model the arch or make assumptions, as those are stated in the problem.
- In Mathematics III, Topic 6, Assessment, Constructed Response, students model when two water tanks hold the same amount of water. The questions guide students on how to solve the problem and the specific information they need to determine the solution. Students do not validate the accuracy of their model.
- In Mathematics III, Topic 10, Constructed Response, page 1, students find the domain and range of a given function and relate them to the context (wind speed) in which the function is presented. Students answer other questions using the function, but students do not develop the model.
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 Agile Mind Integrated series meet expectations for, when used as designed, spending the majority of time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers (WAPs). (Those standards that were not fully attended to by the materials, as noted in indicator 1a.i, are not mentioned here.)
Within Mathematics I, Mathematics II, and Mathematics III, students spend the majority of their time engaging with the WAPs from N-RN, N-Q, A-CED, A-SSE, F-BF, F-IF, F-LE, G-CO, G-SRT, S-ID, and S-IC. Examples of students engaging with the WAPs across the series include:
- Mathematics I: In Topic 5, students explore linear functions through multiple representations (F-IF, F-BF, F-LE, and S-ID). In Topic 7, students explore measures of central tendency and bivariate categorical data (S-ID and S-IC). In Topic 12, students explore exponential growth and decay, compare linear and exponential functions, and create multiple representations of exponential functions (A-CED, A-SSE, F-IF, and F-BF). In Topic 15, students explore rigid transformations and applications of rigid transformations (G-CO and G-SRT).
- Mathematics II: In Topic 1, students explore multiple representations of and solving problems with absolute value and piecewise functions such as the problem of representing shipping charges as a function of the order total (A-CED and F-IF). In Topic 6, students explore real world scenarios leading to quadratic functions, and they solve problems using quadratic equations such as determining how long it will take for a tomato to hit the ground (F-BF and F-LE). In Topic 13, students explore dilations and applications of dilations within the coordinate plane and problems related to enlarging photos. The topic also has students explore how they can prove triangles are similar (G-CO and G-SRT). In Topic 18, students explore regression, modeling exponential growth and decay, and quadratic relationships in real-world scenarios, such as school enrollment and throwing a basketball (N-Q, F-LE, and S-ID).
- Mathematics III: In Topic 1, students explore the similarities and differences between arithmetic, geometric, and infinite series and apply this understanding to problems of a growing triangle pattern or tiling around a fish pond. (A-SSE , F-BF, and F-LE). In Topics 2 and 3, students use probability to make predictions and understand the likelihood of real world events. Students examine different types of statistical studies and the conclusions one can draw from those studies (S-IC). In Topic 10, students examine square and cube root functions, their inverses, and multiple representations of these functions (A-CED and F-IF). In Topic 12, students explore many different real world scenarios of exponential growth and decay and solve problems using equations, graphing, and fitting a function to given data such as the population of fire ants (F-IF and F-LE).
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for Agile Mind Integrated series, when used as designed, meet the expectation for allowing students to fully learn each non-plus standard. Overall, there are multiple opportunities for students to fully learn the non-plus standards by engaging with all aspects of the standards and not distracting students with prerequisite or additional topics.
Examples of the standards where students have multiple opportunities to fully learn the standards include, but are not limited to:
- F-BF.3: Students work with linear functions to identify the effects of transformations in Mathematics I, Topic 6, Student Activity Sheet 4. Students identify the effects of transformations on exponential functions in Mathematics I, Topic 12, Student Activity Sheet 3. Later, students identify the effects of transformation on quadratic and cubic functions in Mathematics III, Topic 4 and rational functions in Mathematics III, Topic 8.
- A-SSE.3a: In Mathematics III, Topic 6, Exploring Quadratics, and in Mathematics II, Topic 4, students factor a quadratic expression to reveal the zeros of the function.
- G-GMD.3: Students use volume formulas to solve problems involving prisms and cylinders in the context of sugar cubes in Mathematics II, Topic 25. In the same topic, students complete a MARS Task related to the volume of a swimming pool. In Mathematics II, Topic 26, students use volume formulas for pyramids and cones in real-world application problems. In Mathematics II, Topic 27, students apply volume formulas to real-world application problems for spherical objects.
- S-ID.1: In Mathematics I, Topic 7, Student Activity Sheet 1, students compare a bar graph and a histogram. Later in the worksheet, students construct a graphical representation of their choice to represent data. In Student Activity Sheet 3 of the same topic, students construct two histograms.
Examples of where the materials do not enable students to fully learn the non-plus standards are:
- F-IF.8b: The materials include exponential growth and decay in Mathematics II, Topic 18, Exploring Modeling Growth and Exploring Modeling Decay. Exponential functions are also addressed in Mathematics III, Topic 12. Students interpret expressions for exponential functions in a limited number of problems.
- F-IF.9: Students compare different functions throughout the materials (e.g. Mathematics II, Topic 2, MARS task: Graphs), but these comparisons include two functions expressed in similar forms (e.g. algebraically, graphically, numerically in tables, etc.). The materials provide a limited number of problems in which students compare functions which are expressed in different forms.
- G-GPE.2: The materials give a geometric definition of a parabola in Mathematics III, Topic 21, Exploring Defining Hyperbolas and Parabolas. The focus and directrix are defined, but the relationship between the focus, directrix, and equation for the parabola is not derived by the students.
- S-IC.6: The materials address reporting conclusions in Mathematics III, Topic 3, Exploring Experiments, but students do not read reports and evaluate them.
- S-CP.4: Students use two-way frequency tables in a variety of problems in Mathematics II, Topic 31, Exploring Conditional Probability, but they do not construct two-way frequency tables.
Indicator 1c
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed for Agile Mind Integrated series meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts, apply key takeaways from Grades 6-8, and vary the types of real numbers being used.
The materials use age appropriate and relevant contexts throughout the series. The following examples illustrate appropriate contexts for high school students:
- In Mathematics I, Topic I, students construct graphs related to a student and his teammates ordering pizza.
- In Mathematics I, Topic 5, students explore the relationship between the depth of gas and number of gallons in a gas tank.
- In Mathematics I, Topic 8, students relate how many snacks you buy with the cost of the snacks at a movie theater.
- In Mathematics II, Topic 19, students model the spread of a flu virus with exponential tables and graphs.
- In Mathematics III, Topic 7, students examine a band selling candy as a fundraiser.
The following problems represent the application of key takeaways from Grades 6-8:
- In Mathematics I, Topic 6, Exploring Transformations, students make connections between geometric transformations and linear equations.
- In Mathematics I, Topic 8, students extend solving linear equations to generate formulas for linear functions and solve linear inequalities.
- In Mathematics I, Topic 9, students use their knowledge of linear functions to build absolute value functions.
- In Mathematics II, Topic 14, students apply proportional reasoning to dilations with a scale factor.
Examples of the materials varying the types of real numbers used across the courses of the series include:
- In Mathematics I, Topic 6, Exploring “Rate of Change”, students perform calculations with decimals as they analyze data from a simulation of Hooke’s Law to create a linear model for the data.
- In Mathematics I, Topic 19, Constructed responses 1 and 2, students perform calculations with decimals as they use geometry to model and solve real-world problems.
- In Mathematics II, Topic 15, Student Activity Sheet 2, Problems 8, 9, 14, 23, and 25, students operate on fractions as they apply laws of exponents to simplify expressions involving rational exponents.
- In Mathematics II, Topic 24, Guided practice, Page 9, work with irrational numbers to solve an area problem.
- In Mathematics III, Topic 1, two problems in the More and Guided Practice sections use a fractional difference in a geometric series.
- In Mathematics III, Topic 7, More practice, Pages 9 and 10, students solve problems about joint variation involving mixed numbers and decimals.
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 Agile Mind Integrated series meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series. Overall, connections between and across multiple standards are made in meaningful ways. Each topic provides a Prerequisite Skills list and an overview of the topic in Topic at a Glance. The Topic at a Glance provides generic connections within each course and throughout the series.
Examples where the materials foster coherence within courses include the following:
- In Mathematics I, students explore linear equations (F-BF.1), graphs, tables, stories, rates of change (F-LE.1b), and key features of the equations and graphs (F-IF.9) in Topics 2-5. In Topic 6, students build on that understanding by modeling linear data with trend lines (S-ID.7), scatterplots, and lines of best fit (S-ID.6a,b,c).
- In Mathematics III, Topic 12, students model data with exponential functions (F-IF.B) and then build on that understanding to construct the concepts of inverse functions and logarithms in Topics 13-15.
- In Mathematics II, Topic 13, Dilations and Similarity begins by connecting the idea of transformations (G-CO.2) to deciding if two triangles are similar (G-SRT.2). This topic uses the properties of similarity to prove AA congruence (G-SRT.2) as well as congruence and similarity criteria to solve problems (G-SRT.5). At the end of this topic, students also prove that all circles are similar (G-C.1).
Examples where the materials foster coherence between the courses include the following:
- In Mathematics I, Topic 13, students explore and create arithmetic and geometric sequences (F-BF.1,2). In Mathematics III, Topic 1, students review arithmetic and geometric sequences to connect this knowledge to arithmetic, geometric, and infinite series (F-IF.3).
- The F-LE standards are connected throughout the series. In Mathematics I, students compare linear growth and exponential growth in a number of topics (Topics 5, 6, and 12). Students use the idea of linear growth in Mathematics II, Topic 10 to find lines that are parallel. In Mathematics III, Topic 14, students use concepts from Mathematics I to work with logarithms using prerequisite knowledge of exponential functions.
- In Mathematics II, Topics 4, 5, and 6, students work with quadratic equations (A-REI.4). Students fit quadratic functions to models in Topic 6 (F-IF.4). Students use their knowledge of quadratic equations from Mathematics II, to explore, factor, and solve polynomial equations in Mathematics III, Topic 6 (A-APR.B).
Indicator 1e
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed for Agile Mind 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 in either the teacher or student materials.
The following are examples of where the materials build on, but do not explicitly identify, standards from Grades 6-8:
- In Mathematics I, Topic 4, Exploring, Constant Rates, students use derived units (e.g. gallons/hour, dollars/cap, bad jeans/total jeans) while building on ratios (6.RP.1).
- In Mathematics I, Topic 15, Exploring, Applications of rigid transformations, students formalize that rotations produce congruent figures. This builds on their understanding of congruence (8.G.2).
- In Mathematics II, Topic 5, Exploring, Complex Numbers, students build on their understanding of the real number system (8.NS.1) to develop an understanding of complex numbers.
- In Mathematics II, Topic 13, Exploring, Dilations, students explore the ratios of side lengths on dilated triangles building on their understanding of similarity (8.G.4).
- In Mathematics II, Topic 19, Exploring, Trigonometric Ratios, the materials develop the trigonometric ratios from proportional reasoning (7.RP.2).
- In Mathematics III, Topic 16, Exploring, Sine and Cosine, students explore “an interesting relationship between cos and sin and use the animation to relate the Pythagorean Theorem to sin and cos to this right triangle.” Students develop the Pythagorean identity from the Pythagorean theorem (8.G.B).
- In Mathematics III, Topic 2, Exploring, Normal Distribution, students learn about the normal distribution by extending their understanding of center and spread (6.SP.2).
Indicator 1f
The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.