High School - Gateway 1
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Focus & Coherence
| Score | |
|---|---|
Gateway 1 - Meets Expectations | 83% |
Criterion 1.1: Focus & Coherence | 15 / 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 High School CPM Integrated series meet the expectation for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses. Overall, the instructional series attends to the full intent of the non-plus standards and the modeling process, spends a majority of time on the widely applicable prerequisites from the CCSSM, and requires students to engage at a level of sophistication appropriate to high school.
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 the High School CPM Integrated series meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The series is designed to spiral. Overall, all of the standards are addressed within the Integrated I, Integrated II, and Integrated III courses.
Most standards are addressed to the full intent throughout the entire series. The following are examples of standards addressed in the series:
- A-APR.1 is addressed in Integrated II and Integrated III. In Integrated II, Lesson 4.1.4 provides evidence explaining how integers are closed under the operations of addition, subtraction, and multiplication; Lesson 4.1.5 has students determine whether polynomials are also closed under these operations by creating several examples to support this notion and then generalize beyond the examples. In Integrated III, Lesson 8.3.1 has students divide polynomials.
- A-REI.3 is addressed in Integrated I. Lesson 3.3.1 utilizes several methods for solving linear equations: rewriting, undoing, and looking inside. Also in Integrated I, Lessons 9.1.1 and 9.1.2 have students solve linear inequalities.
The Number and Quantity domain occurred throughout Integrated I and Integrated II; however, N-Q.1 was only partially addressed. Students were not explicitly expected to use units as a guide to solve problems. Also, students were not required to interpret the origin in every data display.
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 High School CPM Integrated series meet the expectation that the materials attend to the full intent of the modeling process when applied to the modeling standards. The series includes modeling tasks throughout the materials. Frequently, tasks include significant scaffolding or support to focus students on specific mathematics, but scaffolding of modeling tasks decreases within a course and over the series helping to develop students' abilities to work with modeling tasks. In the series, students have opportunities to develop their own solution strategies, select the best tools for solving a problem or set of problems, create their own charts, graphs, and/or equations, evaluate and revise answers, and report on their work.
- In Lesson 2.1.3 of Integrated I, the problem “How Steep Is It?” has students use a model, stairs, to represent the slope of a function. This problem asks students to make, use, and describe a model, but it does not engage them in the full modeling process as defined by the Modeling, High School Progressions Document. This problem addresses standards F-IF.4, F-IF.6, F-IF.7a, F-LE.1a, F-LE.2 and F-LE.5.
- In Integrated I, “The Big Race - Finals” problem in Lesson 2.2.3 has students engage in parts of the modeling process, such as defining variables, interpreting data, validating their conclusions, and reporting out to other teams. However, the students do not come to develop the question themselves, and they do not collect the data for the investigation themselves. All of this is explicitly given to them by the materials. This problem addresses standards N-Q.2, A-CED.2, F-IF.4, F-IF.7a, F-BF.1a, F-LE.1b, F-LE.2 and F-LE.5.
- In “The Burning Candle” problem in Lesson 11.2.3 of Integrated I, students are presented a real-life situation that should be engaging for Integrated I students. The students are asked to design an experiment, collect data and analyze data in order to predict how long a birthday candle will stay lit. The problem appropriately engages students in all aspects of the modeling process. This problem effectively engages students in applying the modeling process to standards N-Q.2, N-Q.3, A-CED.1, F-IF.7a, F-BF.1a, F-LE.2, S-ID.6a, S-ID.6c, S-ID.7 and S-ID.8
- In Integrated II, Lesson 1.2.1 uses a bracelet task to have students perform an experiment and then collect, record and analyze data. Next, they are prompted to modify the experiment, collect, record and analyze new data, and compare the new data set to their first set of results. Finally, the students are prompted to design their own experiment “spin off” of the original and then collect, record and analyze their data. If the lesson is followed through to the end, every aspect of the modeling process would be completed by the students. Right below this task is a flowchart example of modeling with mathematics.
- In Integrated II, “Standards to Maintain” the “Shrinking Targets Lab” in Lesson 9.1.1 has students define variables, collect and analyze data and then use their data to extrapolate. They are provided a significant level of support, but they are still actively engaged in the modeling process. This lab addresses standards A-CED.2, F-IF.4, F-IF.5, F-IF.7a and F-BF.1a.
- In Integrated III, Lesson 6.1.1 has students use coins to model whether a child is born a girl or a boy. The students design the experiment and then record and analyze their results. They are given many parameters that prevent students from determining their own variables. The students share and compare their data with other teams, and they also compile their data and analyze if/how the data changes when the sample size is larger. This lesson addresses standards S-IC.2 and S-MD.6(+).
- In Integrated III, Lesson 6.2.1 has students design a computer simulation to model a real-life situation and then collect and analyze data. This lesson addresses standards S-IC.1, S-IC.4 and S-IC.5.
- In Integrated III, Lesson 9.1.1 (F-TF.5) introduces a task entitled "Emergency!" The experiment procedure is outlined in the materials. The specific directions provided allow for students to focus on the appropriate mathematics and do not detract from the modeling process. Students are asked guiding questions that require them to develop their own strategies for solving the problem and reflect on the difference between their process and the process of others.
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 materials, when used as designed, meet the expectation for allowing 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 (WAPs).
The materials show a strong focus on widely applicable prerequisites.
- The majority of the lessons in Integrated I focus on the WAPs. Chapters 1, 2, 3, 5, 6, 7, 8, 9 and 11 had lessons in which the majority of the time was spent on the WAPs. There are some lessons that review middle school mathematics standards, but this does not occur in a way that is distracting or in a way that takes time away from the WAPs. For example Section 1.3 is projected to last approximately two days and reviews rewriting expression with integer exponents (8.EE.1). Also, Appendix A provides review activities for rewriting expressions (7.EE.1). These sections provide opportunities to support struggling learners and clear up misconceptions but could easily be omitted if not needed by the students.
- The majority of the lessons in Integrated II are spent on the WAPs. Chapters 1, 2, 3, 4, 5, 6, 7 and 9 included lessons which were primarily focused on the WAPs. Chapters 8, 10, 11 and 12 are not focused on the WAPs, but their content does fit the flow of content through the materials. Section 1 of Chapter 1 spends time reviewing attributes of polygons which aligns to middle school standards.
- In Integrated III, approximately half of the lessons focus on WAP standards. The progression and flow of the materials are logical and support a deep understanding of the mathematics. Chapters 1, 2, 3, 5 and 7 included lessons in which the majority of the work was related to the WAPs.
Overall, the majority of student time is spent on the widely applicable prerequisites.
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The materials, when used as designed, partially meet the expectation that students are provided with opportunities to fully learn each non-plus standard. Overall, the lessons are structured in a way that students will fully learn all aspects of most standards and do not distract students with prerequisite or additional topics. However, there are a few missed opportunities for students to make every connection and fully learn all aspects of every standard.
- A-SSE.3b: There are several instances where the student is asked to complete the square and to find the vertex. For example, in Integrated II, Lesson 9.1.3 has students complete the square to find the vertex, but there is no mention of maximum or minimum.
- A-REI.4a: Problem 6-79 in Integrated II outlines the derivation of the Quadratic Formula by completing the square. The materials suggest students should follow the algebraic steps and explain how each step is justified.
- A-REI.7: In Integrated II, Lesson 10.1.1 has one example where students find the intersection between a line and a circle. The intersection is found graphically, not algebraically. In Integrated III, Lesson 3.1.3 has two examples in which students graph a system consisting of a parabola and a circle.
- G-CO.2: Although Lessons 3.1.1-3.1.6 from Integrated I are aligned to this standard, there is no explicit instruction about functions that take points in the plane as inputs and give points as outputs, but there are problems for students to solve in the homework (8-49 and 9-63 from Integrated I).
- G-CO.13: Lessons and problems provide limited opportunities for students to construct a square inscribed in a circle.
- G-GPE.6: There were a limited number of problems that partitioned segments in a ratio other than 1:1.
- F-IF.2: Students use function notation and evaluate functions for the inputs in their domains. Problems requiring students to interpret statements that use function notation in terms of a context such as problem 2-53 in Integrated I are infrequent.
- F-IF.6: There was limited evidence found of finding the average rate of change in non-linear situations. In Integrated II, Lessons 9.3.1 and 9.3.2 and problems 9-73 and 9-74 have students work with parabolas. In problem 9-94, students work with linear, quadratic and exponential situations. These questions and activities have students calculate and interpret the average rate of change of a function over a specified interval. These problems do not require estimation or interpretation. In Integrated III, lesson 2.2.5 and problem 2-111 have students work with non-linear situations with graphs and tables and require them to do some interpretation.
- F-IF.4 and F-IF.7c: Key features are interpreted throughout the series; however, there was little evidence of the actual verbiage of "end behavior" used.
Indicator 1c
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed fully meet the expectations for students to engage in mathematics at a level of sophistication appropriate to high school. The materials give extensive opportunities to work with course-level problems and exercises appropriate to high school and relate new concepts to students' prior skills and knowledge.
The Universal Access section allows all students an opportunity for entry points to the content. There is appropriate guidance for the teacher to help scaffold for different students, and the level of scaffolding and support is appropriate and does not impede students from engaging in the full intent of the mathematics.
Contextual problems are appropriate for high school students. Several contextual problems complement content that students learn in other core classes, such as farming and sustainability, exercise, and genetics.
Scenarios presented in application problems are authentic, as well as adjustable to different interests. Examples of authentic application and/or real world problems include the following:
- In Integrated I, Lesson 1.1.2 uses three contextual scenarios, placement of tiles in a yard, modeling the spread of a flu epidemic, and time it takes to sign your name on multiple documents, in order to teach growth of patterns.
- In Integrated II, Lesson 3.1.3 uses a rock-paper-scissors game with scoring rules which may or may not be fair. Students have to decide the fairness based on their knowledge of probability models.
- In Integrated III, Lesson 6.2.1 focuses on statistical testing using sampling variability. The lesson poses a question of whether students support keeping or canceling a winter formal dance.
Students work with appropriate numbers for high school and see a wide variety in equation/expression formats.
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 meet the expectations that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and required by the standards.
All conceptual categories are addressed over each of the courses. Each course contains work in number and quantity, algebra, functions, geometry, and statistics and probability. Topics are addressed when they are developmentally appropriate. More concrete ideas are examined in Integrated I while more abstract ideas are examined in Integrated III. The progression of difficulty is logical.
The following are examples of connections made between the books in the series:
- Connections are made throughout every course in the Review & Preview portion of every Section. These problems connect with prior work in both the current course and past courses (if any), the current topic, and future topics (usually using the preview problems to review skills and concepts for work that is immediately upcoming).
- In Integrated II, Chapter 2 provides students the opportunity to make connections from their work on congruence in Integrated I with a brief review of congruence theorems to the work that they will do with similarity and dilations.
The following are examples of connections made within the books in the series:
- In Integrated I, Chapter 7 starts by engaging students in what it means for two figures to be congruent, and then it engages in determining the least amount of information needed for proving two figures to be congruent. It proves triangle congruence criteria using rigid motions. Then it has the opportunity to connect that work to the coordinate plane. The students study polygons on the coordinate grid by proving statements about the figures using coordinate geometry and relationships for distance, slope, and midpoint.
- In Integrated II Chapter 9, Modeling with Functions, the study of quadratic associations in statistics and probability builds on students' understanding of quadratic relationships, from Chapters 2, 5 and 6.
The materials are designed to spiral concepts throughout the chapters and courses. Some topics included within the same chapter are disconnected. These were placed this way intentionally to allow students more time with the first concepts in Review & Preview before the concept is developed further in a future chapter.
- In Integrated III, Chapter 7 is on logarithms and triangles. The connection between logarithms and triangles is not evident.
- In Integrated II, Chapter 7 is about Proof and Conditional Probability, and Chapter 3 is about Probability and Trigonometry. The connection between these topics is not evident.
Indicator 1e
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed partially meet the expectations that the series explicitly identify and build on knowledge from Grades 6-8 to the High School Standards. The content that is identified as being from previous grades is appropriate and develops as a natural progression into high school, but it is not always clearly connected to a specific middle school standard.
The following are examples of connections made between content from Grades 6-8 and the high school content:
Integrated I:
- Chapter 1 begins with a comment that "in previous courses you may have learned about relationships between two quantities that could be graphed using a straight line." It goes on to discuss what will be learned in this chapter. Also, the Overview of the Chapter states that Section 1.3 should be a review from Grade 8. Students are reminded that "in previous courses, you might have looked at patterns, in tables, graphs, equations, and situations that were linear."
- Before Chapter 2, it is highly recommended to look through Appendix A to determine if the students need the additional lessons before continuing. In the overview of the chapter for Chapter 2, the Teacher Edition states that the chapter will build on the study of linear functions from previous courses. Chapter 2 is mostly middle grades work that examines patterns of growth, slope, and writing and using linear equations that aligns to 8.F.4 and 8.F.5, with a few exceptions and without being identified. Lesson 2.2.4 teaches unit conversion with ratios (6.RP.3d). As students progress through Chapter 2, connections are made to F-BF.1a, F-LE.1a,b and N-Q.1,2.
- Lesson 3.1.1 uses the notions of reflections, rotations, translations (8.G.1), and nets (6.G.4) to introduce ideas of visualization in three dimensions (G-GMD.4) but does not identify the middle grade standard.
- In Chapter 3, Lesson 3.2.1 mentions that in previous courses, students worked with areas and perimeters of shapes made up of different rectangles.
- The Overview of the Chapter for Chapter 7 states that students start the chapter by reviewing what they know from previous courses about similarity. There is also a Math Notes box that reviews similar triangles and scale factor.
- The teacher's notes for Lesson 10.1.2 state that, in previous courses, students will have computed mean and five-number summaries and will have described the shape of a distribution. They will have also chosen between mean and median based on the shape of the distribution and will have calculated the mean absolute deviation and interquartile range.
- Appendix A is appropriately identified as middle school content. Specific standards are not presented.
Integrated II:
- In Chapter 1, Lesson 1.1.1 states that in previous courses students have studied polygons like triangles and quadrilaterals. Lessons 1.1.1-1.1.2 have students composing polygons and examining attributes of polygons. The material is reminiscent of various elementary standards, such as 3.G.1, 4.G.2 and 5.G.3. No standards of any level are identified for these lessons. Lesson 1.3.1 reviews angle pairs (complementary, supplementary, vertical, linear pairs - 7.G.5). The work supports the high school work but is not identified as middle school. Lessons 1.3.2 and 1.3.3 do the same with angles formed by parallel lines intersected by a transversal (8.G.5).
- The Chapter 3 Overview states that students will be expanding their understanding of simple probability studied in middle school. Lesson 3.1.1 states that in previous courses students studied probability.
- In Chapter 5, Lesson 5.1.1 states that in previous courses students have investigated linear and exponential functions.
- In Chapter 6, Lesson 6.1.4 states that students learned about the laws of exponents in a previous course.
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.
The instructional materials reviewed for the High School CPM Integrated series explicitly identify the plus standards, when included, and the plus standards coherently support the mathematics which all students should study in order to be college- and career-ready.
In most cases, it would be difficult to separate out the plus standards material from the non-plus standards. However, work with the plus standards does not deter from the work with the non-plus standards. An example of where it could easily be separated is in Integrated II, Chapter 12, which is comprised of primarily plus standards; this section could be omitted. The materials, however, offer no guidance or pacing suggestions should teachers not wish to use the plus standards.
The plus standards that are identified in the teacher resource binder and addressed to reach the full intent of the standard are listed below:
- Integrated I does not address any plus standards.
- Integrated II: N-CN.8 (Lessons 5.2.6 and 6.2.4); N-CN.9 (Lesson 6.2.4); G-C.4 (Lesson 10.2.5); S-CP.8 (Lesson 7.2.3); S-CP.9 (Lessons 12.1.1-12.1.4 and 12.2.4); S-MD.6 (Lessons 3.1.5, 12.1.1-12.1.3 and 12.2.4); S-MD.7 (Lessons 6.2.6, 7.2.3, 12.1.1, 12.2.1 and 12.2.4)
- Integrated III: N-CN.8 (Lessons 8.2.2 and 8.3.4); N-CN.9 (Lessons 8.1.1, 8.1.2, 8.2.1, 8.2.2, and 8.3.2); A-APR.5 (Lessons 10.3.1-10.3.3); A-APR.7 (Lessons 11.1-11.1.4 ); F-TF.9 (Lessons 12.2.1, 12.2.2 and 12.2.3); G-SRT.9 (Lessons 7.2.1 and 7.2.2); G-SRT.10 (Lessons 7.2.1-7.2.4); G-SRT.11 (Lesson 7.2.5); S-MD.6 (Lessons 6.1.1, 6.1.2 and 6.3.1); S-MD.7 (Lesson 6.2.3)
One plus standard is assessed on checkpoints. Standard N-CN.8 is assessed on Integrated III Checkpoint 11.