High School - Gateway 3

The materials reviewed for Fishtank Math AGA partially meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the students material and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The materials provide some general guidance that will assist teachers in presenting the students and ancillary materials, but they do not consistently include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Examples include, but are not limited to:

• The Algebra 1 and Algebra 2 course summary includes a section which states, “How do we order the units?  In Unit 1...in Unit 2..” which provides a synopsis of the work the students will be engaging in. However, the Geometry course summary does not offer such an explanation.

• In most lessons, solutions are not provided for Anchor Problems and/or Target Tasks. In a few lessons, Anchor Problem(s) and/or Target Task(s) solutions are available through a link to the source of the problem. For example, Geometry, Unit 8, Lesson 1, the solutions available for the Anchor Problems through links to the sources. 

• Tips for Teachers provides strategies and guidance for lesson implementation; however, there are several lessons that contain no Tips for Teachers. Examples include, but are not limited to: Algebra 1, Unit 2, Lessons 8, 15, 18, Algebra 2, Unit 5, Lessons 10 - 13, and Geometry, Unit 3, Lesson 1 - 6, 8, 11, and 14 - 17.

The materials provide minimal guidance that might assist teachers in presenting the ancillary materials. Examples include:

• The Preparing to Teach a Math Unit section, gives seven steps for teachers to prepare to teach a unit, as well as questions teachers should ask themselves when organizing a lesson presentation. For example, Step 1 states, “Read and annotate the Unit Summary - Ask yourself: What content and strategies will students learn?”, “What knowledge from previous grade levels will students bring to this unit?”, and “How does this unit connect to future units and/or grade levels?”.

• At the beginning of each unit there is a Unit Summary section, which provides a synopsis of the unit, Assessment links, Unit Prep, and identifies Essential Understandings connected to the unit. For example, in Algebra 2, Unit 5, the Unit Summary is about Exponential Modeling and Logarithms. Teachers are informed that students have previously seen exponential functions in Algebra 1, and that this unit builds upon prior knowledge by revisiting exponential functions and including geometric sequences and series and continuous compounding situations. The Unit Prep contains suggestions for how teachers can prepare for unit instruction. This includes suggestions for taking the unit assessment, making sure to note which standards each question aligns to, the purpose of each question, strategies used in the lessons, relationships to the essential questions and lessons that the assessment is aligned to. It is suggested that teachers do all of the target tasks, and make connections to the essential questions and the assessment questions. A vocabulary section tells the teacher the terms and notation that students will learn or use in the unit but does not define them. Additionally, a materials section lists the materials, representations, and/or tools that teachers and students will need for this unit.

The materials reviewed for Fishtank Math AGA partially meet expectations for containing adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Materials contain adult-level explanations and examples of the more complex course-level concepts so that teachers can improve their own knowledge of the subject. While adult-level explanations of concepts beyond the course are not present, Tips for Teachers, within some lessons, can support teachers in developing a deeper understanding of course concepts. Opportunities for teachers to expand their knowledge include:

• In Algebra 1, Unit 1, Lesson 2, the Tips for Teachers, contain adult-level explanations of complex grade level concepts. There is a link to a resource to show teachers all of the ways that function notation can be represented. The linked materials look at functions in a more sophisticated manner - “y is a function of x” what does this mean and what is the relationship between x and y. Lesson 2 is the first lesson that introduces function notation to students.

• In Algebra 2, Unit 1, Lesson 4, the Tips for Teachers, contain adult-level explanations of complex grade level concepts. It states, “This lesson has components that extend beyond F-BF.4a into F-BF.4c and F-BF.4d. If you are not teaching an advanced Algebra 2 course, focus on the contextual meaning of inverse functions presented in this lesson rather than the tabular or graphical analysis of inverse functions. The following resource may be helpful for teachers to grasp the full conceptual understanding of inverse functions before planning this lesson. American Mathematical Society Blogs, Art Duval, “Inverse Functions: We’re Teaching It All Wrong!” November 28, 2016.” In this piece, Duval explains the problems that can occur with switching variables in the sense that the meaning of the variables can change. This understanding of inverse relationships (course-level content) extends beyond the intent of the standards. 

• In Geometry, Unit 4, Lessons 6 and 7, the Tips for Teachers, contain adult-level explanations of complex grade level concepts. It states, “The following resource can help to frame the overall study of introductory trigonometry: Continuous Everywhere but Differentiable Nowhere, Sam Shah, “My Introduction to Trigonometry Unit for Geometry”. The purpose of this blog is to help teachers develop a deeper understanding of the lessons concepts of trigonometry.

The materials reviewed for Fishtank Math AGA partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials identify the standards and practices assessed for some of the formal assessments.

There is a Post-Unit Assessment for each unit in a course. Assessment item types include short-answer, multiple choice, and constructed response. For the Algebra 1 and Geometry courses, there are Post-Unit Assessment Answer Keys for each unit. The Algebra 1 and Geometry Post-Unit Assessment Answer Keys for each unit contains the following: question numbers, aligned standards, item types, point values, correct answers and scoring guides, and aligned aspects of rigor for each question. However, neither the Post-Unit Assessment or the Post-Unit Answer Keys identify the mathematical practice. Examples of Algebra 1 and Geometry Post-Unit Assessment questions and aligned standards include: 

• Algebra 1, Unit 3: For Post-Unit Assessment question 4, students solve a multi-step inequality.The answer key shows the aligned standard as A-REI.3. This is a short-answer question with a point value of 2, and the rubric explains how the two points are determined based on the detailed accuracy of the student’s answer.The aspect of rigor for this question is referenced as P/F (procedural fluency).  

• Algebra 1, Unit 5: For Post-Unit Assessment question 3, students are given a system of equations and the graph of both functions (One is a linear function and the other is an absolute value function). For part 3a, students need to identify the solution(s) for the system of equations; then for part 3b, they need to algebraically show that the point(s) are solution(s) to the system. Each part of this question aligns with standard A-REI.11 and each part has a point value of 2. Part 3a item type is considered as short-answer and part 3b’s item type is identified as constructed response. Aspects of rigor for this question are referenced as C P/F (conceptual understanding/ procedural fluency).

• Geometry, Unit 5: For Post-Unit Assessment question 3, students are given a constructed response task consisting of the graph of a triangle (CAB) and 3 related questions; students need to calculate \frac{2}{3} of the distance between points C and A (part a), and points C and B (part b). Students label these new coordinate points D and E respectively, found by completing these calculations. Students then calculate the perimeter of triangle CDE in radical form (part c). The aligned standard for the first two parts of this question is G-GPE.6 and the aligned standard for the third part is G-GPE.7. A point value of 1 is assigned to each of the first two parts of the question and a point value of 2 is assigned to the third part of the question.

For the Algebra 2 Course, Post-Unit Assessments have no answer keys and there is no alignment of questions to the standards. Examples of Algebra 2 Post-Unit Assessments that have no answer keys or standards referenced include, but are not limited to: Algebra 2, Units 1, 2, 5, and 9. The following Algebra 2 Post-Unit Assessments have some solutions and standards referenced in links to original sources:

• Algebra 2, Unit 3, only have references for questions 8, 9, and 13

• Algebra 2, Unit 4, only have references for questions 2, 4, 5, and 14

• Algebra 2, Unit 7, only have references for questions 1, 2, 4, 6, and bonus question

Many of these reference links do not work, such as for Regents Exams in units 4 and 7 and in “Algebra II Paper-based Practice Test from Mathematics Practice Tests,” from unit 3.

The materials reviewed for Fishtank Math AGA partially meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series.

The Assessments section found under Math Teacher Tools contains the following statement: “Pre-unit and mid-unit assessments as well as lesson-level Target Tasks offer opportunities for teachers to gather information about what students know and don’t know while they are still engaged in the content of the unit. Post-unit assessments offer insights into content that students may need to revisit throughout the rest of the year to ensure continued work towards mastery. Student self-assessments provide space for students to reflect on their learning and monitor their own progress.” The materials reviewed do not contain Pre-Unit, mid-unit, or student self-assessments, the system of assessments included is twofold: Target Tasks and Post-Unit Assessments. All of the Post-Unit Assessments have to be printed and administered in person. For the Algebra 1 and Geometry unit assessments, answer keys are provided, however no answer keys are provided for the Algebra 2 unit assessments.The unit assessment item types include multiple choice, short answer, and constructed response. However, the assessment system leaves standards unassessed.

Examples of how standards are not assessed or only partially assessed in Post-Unit Assessments include, but are not limited to:

• In Algebra 1, Unit 3, students solve equations and inequalities. However, students are not prompted to explain each step. ( A-REI.1)

• In Algebra 1, Unit 8, students identify the vertex, minimum or maximum, axis of symmetry, and y-intercept, but they do not indicate where the function is increasing or decreasing or whether the function is positive or negative (F-IF.4).

• In Algebra 2, Unit 8, there are several questions that involve random sampling, but students do not explain how randomization relates to the context.

• In Geometry, Unit 1, students construct an equilateral triangle by copying a segment (G-CO.12). G-CO.13 is identified as being addressed in the same unit, but it is not assessed in the Post-Unit assessment.

In Post-Unit Assessment Keys for Algebra 1 and Geometry, Common Core Standards are identified for each assessment item, but mathematical practices are not identified for any of the assessment items. Examples of Post-Unit Assessment multiple choice items include:

• Algebra 2, Unit 2, question 2 asks, “Which equation has non-real solutions? a. 2x^2+4x-12=0  b. 2x^2+3x=4x+12 c. 2x^2+4x+12=0  d. 2x^2+4x=0“

• Geometry, Unit 7, question 1 states, “A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of 10 cm. What is the largest number of necklaces that can be made from 1000 cm of wire? A. 16, B. 15, C. 31  D. 32.”

Examples of Post-Unit Assessment short answer items include:

• Algebra 1, Unit 6, question 5 states, “Multiply and simplify as much as possible: \sqrt{8x^3 \cdot \sqrt{50x}} “

• Algebra 2, Unit 1, question 1 states, “Let the function f  be defined as f(x)=2x+3a  where a is a constant. a. If f(-5)= -4  , what is the value of the y-intercept? b. The point (5,k) lies of the line of the function f . What is the value of k?”

Examples of Post-Unit Assessment constructed response items include:

• Algebra 1, Unit 5, question 6 states, “A new small company wants to order business cards with its logo and information to help spread the word of their business. One website offers different rates depending on how many cards are ordered. If you order 100 or fewer cards, the rate is $0.40 per card. If you order over 100 and up to and including 200 cards, the rate is $0.36 per card. If you order over 200 and up to and including 500 cards, the rate is $0.32 per card. Finally, if you order over 500 cards, the rate is $0.29 per card.

  • Part A: Write a piecewise function, p(x), to model the pricing policy of the website.

  • Part B: Calculate p(250-p(200), and interpret its meaning in context of the situation.

  • Part C: The manager of the company decides to order 500 business cards, but the marketing director says they can order more cards for less money. Is the marketing director’s claim true? Explain and justify your response using calculations from the piecewise function.”

• Algebra 2, Unit 8, question 9 states, “A brown paper bag has five cubes, 2 red and 3 yellow. A cube is chosen from the bag and put on the table, and then another cube is taken from the bag. 

  • Part A: What is the probability of two red cubes being chosen in a row? 

  • Part B: Is the event of choosing a red cube the first time you pick and choosing a red cube the second time you pick from the bag independent events? Why or why not?”

The materials reviewed for Fishtank Math AGA partially meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through active participation in grade-level mathematics, but not consistently.

While there are resources within Math Teacher Tools, Supporting English Learners, that provide teachers with strategies and supports to help English Learners meet grade-level mathematics, these strategies and supports are not consistently embedded within lessons. The materials state, “Our core content provides a solid foundation for prompting English language development, but English learners need additional scaffolds and supports in order to develop English proficiency as they build their content knowledge. In this resource we have outlined the process our teachers use to internalize units and lessons to support the needs of English learners, as well as three major strategies that can help support English learners in all classrooms (scaffolds, oral language protocols, and graphic organizers). We have also included suggestions for how to use these strategies to provide both light and heavy support to English learners. We believe the decision of which supports are needed is best made by teachers, who know their students English proficiency levels best. Since each state uses different scales of measurement to determine students’ level of language proficiency, teachers should refer to these scales to determine if a student needs light or heavy support. For example, at Match we use the WIDA ELD levels; students who are levels 3-6 most often benefit from light support, while students who are levels 1-3 benefit from heavy support.”Regular and active participation of students who read, write, and/or speak in a language other than English is not consistently supported because specific recommendations are not connected to daily learning objectives, standards, and/or tasks within grade-level lessons. 

Within Supporting English Learners there are four sections, however only one section, Planning for English Learners, is available in Fishtank Math AGA. Planning for English Learners is divided into two sections, Intellectually Preparing a Unit and Intellectually Preparing a Lesson.

• The “Intellectually Preparing a Unit” section states, “Teachers need a deep understanding of the language and content demands and goals of a unit in order to create a strategic plan for how to support students, especially English learners, over the course of the unit. We encourage all teachers working with English learners to use the following process to prepare to teach each unit.” The process is divided into four steps where teachers are prompted to ask themselves a series of questions such as: “What makes the task linguistically complex?”, “What are the overall language goals for the unit?”, and “What might be new or unfamiliar to students about this particular mathematical context?”

• The “Intellectually Preparing a Lesson” section states, “We believe that teacher intellectual preparation, specifically internalizing daily lesson plans, is a key component of student learning and growth. Teachers need to deeply know the content and create a plan for how to support students, especially English learners, to ensure mastery. Teachers know the needs of the students in their classroom better than anyone else, therefore, they should also make decisions about where to scaffold or include additional supports for English learners. We encourage all teachers working with English learners to use the following process to prepare to teach a lesson.” The process is divided into two steps where teachers are prompted to do certain objectives such as , “Unpack the Objective, Target Task, and Criteria for Success” or “Internalize the Mastery Response to the Target Task” or to ask themselves a series of questions such as: “What does a mastery answer look like?”, “What are the language demands of the particular task?” , and “If students don't understand something, is there a strategy or way you can show them how to break it down?”

Regular and active participation of students who read, write, and/or speak in a language other than English is not consistently supported because specific recommendations are not connected to daily learning objectives, standards, and/or tasks within grade-level lessons.