Alignment: Overall Summary

The materials reviewed for Fishtank Math AGA meet expectations for alignment to the CCSSM for high school. For focus and coherence, the series showed strengths in the following areas: attending to the full intent of the mathematical content contained in the standards, spending the majority of time on the content from CCSSM widely applicable as prerequisites, requiring students to engage in mathematics at a level of sophistication appropriate to high school, being mathematically coherent and making meaningful connections in a single course and throughout the series, and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. In Gateway 2, for rigor, the series showed strengths in the following areas: supporting the intentional development of students' conceptual understanding, opportunities for students to develop procedural skills, working with applications, and displaying a balance among the three aspects of rigor. The materials intentionally develop all of the eight mathematical practices, but do not explicitly identify them in the context of individual lessons.

Alignment

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Meets Expectations

Gateway 1:

Focus & Coherence

0
9
14
18
15
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
9
14
16
15
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Usability

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Does Not Meet Expectations

Not Rated

Gateway 3:

Usability

0
17
24
27
15
24-27
Meets Expectations
18-23
Partially Meets Expectations
0-17
Does Not Meet Expectations

Gateway One

Focus & Coherence

Meets Expectations

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Gateway One Details

The materials reviewed for Fishtank Math AGA meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content contained in the standards, spending the majority of time on the content from CCSSM widely applicable as prerequisites, requiring students to engage in mathematics at a level of sophistication appropriate to high school, being mathematically coherent and making meaningful connections in a single course and throughout the series, and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The materials partially meet expectations for attending to the full intent of the modeling process and letting students fully learn each non-plus standard.

Criterion 1a - 1f

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”.

15/18
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content contained in the standards, spending the majority of time on the content from CCSSM widely applicable as prerequisites, requiring students to engage in mathematics at a level of sophistication appropriate to high school, being mathematically coherent and making meaningful connections in a single course and throughout the series, and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The materials partially meet expectations for attending to the full intent of the modeling process and letting students fully learn each non-plus standard.

Indicator 1a

Materials focus on the high school standards.

Indicator 1a.i

Materials attend to the full intent of the mathematical content in the high school standards for all students.

4/4
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Examples of standards addressed by the courses of the series include:

  • N-Q.1: In Algebra 1, Unit 2, Lesson 15, Anchor Problem 1, students label axes with appropriate variables and units. Guiding Questions include, “How did you determine an appropriate scale?” In Algebra 2, Unit 1, Lesson 4, students use graphs to interpret units. Guiding Questions include, “How could George use the graph of f(g) to find the number of quarts that equals three-quarters of a gallon?” and “ Which function is more appropriate to use to find the number of quarts that equals a gallon?” In Geometry, Unit 6, Lesson 16, Anchor Problem 2, students use metric unit conversions to solve the problem.

  • N-CN.2: In Algebra 2, Unit 2, Lesson 8, Anchor Problem 1, students add and subtract complex numbers, and to determine if properties of operations apply to the addition and subtraction of complex numbers. In Anchor Problem 2, students find the product of two complex numbers, and determine if properties of operations apply to the multiplication of complex numbers.

  • A-REI.2: In Algebra 2, Unit 4, Lesson 14, Anchor Problem 3, students compare two radical equations graphically to see that there are no solutions, although it may appear there are solutions if students attempt to solve them algebraically. In Lesson 15, Anchor Problem 1, students generate two solutions, but when they test their solutions, they discover one does not work in the original equation.

  • F-IF.7c: In Algebra 2, Unit 3, Lesson 3, Anchor Problem 1, students are given sketches of two different functions and the factored form of one of them. Using Guiding Questions (“What is similar about the graphs?, What is different?,” and “How do these differences help you determine which graph matches the equation?”), students can match the factored form to the correct graph. Other Guiding Questions lead students to determine the end behaviors of the graphs. Within the same lesson, Anchor Problem 2 has students use the roots of a cubic function to sketch the function.

  • G-C.5: In Geometry, Unit 7, Lesson 11, Anchor Problem 2, students use Guiding Questions to develop a conceptual understanding of the proportional relationship between the radius of a circle and the length of an arc. Guiding Questions include, "What would be the ‘arc length’ if you were to measure the entire outside of the circle?” and “What portion of the entire circumference are you measuring (for a 30-degree angle)?” In Lesson 12, Anchor Problem 2, students use Guiding Questions and a diagram of concentric circles to develop a conceptual understanding of the proportional relationship between the radius and arc lengths. Finally, in Lesson 13, Anchor Problem 1, students find the area of a sector using its proportional relationship with the whole circle. With the use of Guiding Questions, students write a general formula to determine the sector area of a circle with respect to its central angle measure and radius length.

  • G-GPE.6: In Geometry, Unit 5, Lesson 2, Anchor Problem 1, students identify locations that would partition a piece of wood into a 3:5 ratio. Students use a number line to justify their answers. In Anchor Problem 2, students use a number line to partition a line segment into a 3:4 ratio. In Anchor Problem 3, students use coordinates on a plane, and partition the vertical line segment into a 1:2 ratio. In Lesson 3, Anchor Problem 1, students find the midpoint of a directed line segment on a coordinate plane. In Anchor Problem 2, students partition a directed line segment into a 1:3 ratio.

  • S-ID.4: In Algebra 1, Unit 2, Lesson 8, students annotate a standard deviation on a curve with correct values.Through calculations, they locate two points between which 68% of the contextual data falls and then calculate a percent of the data falling between two specified points. In Algebra 2, Unit 8, Lesson 8, Anchor Problem 2, students revisit and build a deeper understanding of normal distributions with the use of a contextual problem involving heights of 8 year old boys. Students use the information given in the problem and a graph of the normal distribution to calculate percentages related to the data. In Lesson 9 of the same unit, Anchor Problem 1, students revisit the problem involving the normal distribution of heights of 8 year old boys to calculate z-scores (number of standard deviations that a score is away from the mean). 

  • S-IC.5: Algebra 2, Unit 8, Lesson 13, Anchor Problem 1, students use a dotplot of the difference of means to compare plant growth in standard and nutrient-treated soils. The Guiding Questions include, "What would the distribution of data look like for you to confidently conclude that there is not enough evidence to say the nutrient-treated soil contributed to growth?" TheTarget Task has a histogram that represents another problem involving the difference of means in relation to plant growth in different soils. Students consider whether the difference of the means is due to the way the sample was taken or whether the treatment had an effect.

Indicator 1a.ii

Materials attend to the full intent of the modeling process when applied to the modeling standards.

1/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. In this series, various aspects of the modeling process are present in isolation or combinations, yet opportunities for the complete modeling process are absent for the modeling standards throughout the materials. 

Examples of problems that allow students to engage in some aspects of the modeling process include, but are not limited to:

  • Algebra 1, Unit 2, Lesson 21: An Anchor Problem contains data involving percent pass-completion rates for top paid NFL quarterbacks and their salaries. Students organize the data, represent it visually, and calculate bivariate statistical measures. Students can use Guiding Questions to help them formulate a problem related to the data. Examples of Guiding Questions include: “How did you organize this data set when the values of the salaries are so high?,” “Why did you decide to use the graph you did?,” and “What formulas will you use to either find the measures you need to make the graph or calculate center and spread?” Students make decisions about how to organize and represent the data graphically and ways to use the data. How the data is used will determine what statistical measures they need to calculate. Students can analyze and interpret the data to solve a problem, but there are no explicit instructions for how they should report their findings (N-Q.1, S-ID.6, S-ID.7, S-ID.8, S-ID.9).

  • Algebra 1, Unit 8, Lesson 12, EngageNY Mathematics, Algebra 1, Module 4, Topic B, Lesson 16: “The Exploratory Challenge,” provides a scenario where a fence is being constructed. Students are given the variables to use when writing an expression; thus, not allowing students to formulate their own. Students find the maximum area and determine if their answer is surprising. This allows for students to interpret and validate their response; however, there is no clear way that students should report their answer. (A-CED.2, F-IF.8a). 

  • Algebra 2, Unit 2, Lesson 9: Students focus on quadratic functions in context. The Target Task has two people throw a baseball in the air. One ball is modeled by a function, the other by a graph. One person says his ball goes higher and the students must decide if he is correct, determine how long each ball was in the air, and construct a graph of the function given to back up claims from the first two parts. Students do not have opportunities to formulate the mathematical model, but do have opportunities to validate and interpret their responses in relation to the problem. However, there is no directed way for students to communicate and/or report their findings to others (A-CED.1, A-CED.2, F-IF.6, F-BF.1). 

  • Algebra 2, Unit 6, Lesson 14, Problem Set, EngageNY Mathematics, Algebra II, Module 2, Topic B, Lesson 13: “Tides, Sound Waves, and Stock Markets,” students write a sinusoidal function to fit a set of data. Students manipulate their function as the data will not lie exactly on the graph of the function. Students then analyze their model to predict a later time and height. Reflection questions prompt students to consider variance during different times of the year. There are other, similar modeling questions in the links to afford students opportunities to improve their ability to fit and analyze sinusoidal curves. However, there is no requirement for students to write a report and communicate their findings (F-TF.5). 

  • Geometry, Unit 6, Lesson 15, Anchor Problem 2: Students are given an image of four packages that all contain the same amount of candy, and then asked to rank the packages based on the least amount of packaging used to the most. The Guiding Questions give students a series of questions to consider, then asks them to determine a different configuration that would have a better packaging choice than one of the four presented. Students do not use the Guiding Questions to represent this problem mathematically. Guiding Questions assist students with rationalizing their responses, and developing their thinking on how to know if their answers are reasonable. With the use of the Guiding Questions, students can manipulate the model and validate and interpret their responses in relation to the problem, but there are no explicit instructions for how they should report their findings (G-MG.1).

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.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA, when used as designed, meet expectations for allowing students to spend the majority of their time on the CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, postsecondary programs and careers.

Examples of standards addressed by the series that have students engaging in the WAPs include:

  • N-Q: In Algebra 1, Unit 1, Lesson 1, students model contextual linear data graphically, using appropriate scales and key graph features (N-Q.1). In Geometry, Unit 6, Lesson 2, students calculate and justify composite area and circumference of circles by defining appropriate units and levels of precision of measurement (N-Q.2 and N-Q.3). In Algebra 2, Unit 1, Lesson 4, Anchor Problem 1, students consider appropriate scaling for graphs with respect to units and use graphs to define appropriate quantities related to a problem context (N-Q.1 and N-Q.2). In Algebra 2, Unit 4, Lesson 17, students analyze unit relationships found in context data and associate units with variables to write expressions that model the data (N-Q.1). 

  • A-CED: In Algebra 1, Unit 3, Lesson 6, Anchor Problem, students are given the formula that describes a quantity related to getting to a destination via walking and riding a bus. The Guiding Questions ask how they would solve for one of the variables to determine the meaning of a specific variable (A-CED.4). In addition, they find the units associated with the variable, attending to N-Q.1. Lastly, students determine the domain restrictions that would be placed on different variables in the context of the problem (F-IF.5). In Geometry, Unit 5, Lesson 14, students write a system of inequalities to represent the polygon in a coordinate plane (A-CED.3). In Algebra 2, Unit 1, Lesson 6, Anchor Problem 2, students write and solve a one-variable equation in context (A-CED.1), then use that solution to write a system of equations in two variables and answer questions in context (A-CED.2 and A-CED.3). 

  • F-IF: In Algebra 1, Unit 1, Lesson 1, students calculate the average time per mile for various commutes (F-IF.6). In Algebra 1, Unit 4, Lesson 3, Anchor Problem 1, students calculate the slope of a function using a table of values (F-IF.6), and in Anchor Problem 2, compare properties of two functions represented algebraically and numerically in a table (F-IF.9). In Algebra 1, Unit 4, Lesson 4, Anchor Problems 1 and 2, students associate the domain with inputs as they explore contextual restrictions (F-IF.1), and evaluate functions for inputs and interpret function notation in context (F-IF.2). In Algebra 1, Unit 6, Lessons 11-15, and Algebra 2, Unit 5, Lesson 1, students define and write explicit and recursive formulas for arithmetic and geometric sequences (F-IF3). In Algebra 2, Unit 3, Lessons 1 and 3, students begin graphing polynomials using tables. Students are given a sketch of two different graphs and the factored form of one of the graphs. Students need to match the correct graph to the factored form of the function. Additionally, they identify the end behavior of the function (F-IF.7c,9).

  • G-CO: In Geometry, Unit 1, Lessons 1-5, students define and construct geometric figures using a straightedge and a compass, including: angles and angle bisectors, an equilateral triangle inscribed in a circle, perpendicular bisectors and altitudes of triangles (G-CO.1, G-CO.12 and G-CO.13). In Geometry, Unit 4, Lesson 1 and Unit 6, Lessons 4 and 5, students define parts of the right triangle, describe the terms “point, line, and plane,” define polyhedrons (prisms and pyramids), and define cylinders and cones (G-CO.1). Throughout Geometry Units 1,4 and 6, G-CO.1 is addressed and applied with other WAPs standards (G-SRT.4-8) to develop and use working definitions for angle, circle, perpendicular line, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

  • S-IC.1: In Algebra 1, Unit 2, Lesson 1, students explore graphs to discuss how randomness and statistics are used to make decisions. In Algebra 2, Unit 8, Lesson 7, students use sample results from several trials to make predictions about the shape of a population.

Indicator 1b.ii

Materials when used as designed allow students to fully learn each standard.

2/4
+
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA, when used as designed, partially meet expectations for allowing students to fully learn each standard. Examples of the non-plus standards that would not be fully learned by students include:

  • N-RN.3: In Algebra 1, Unit 6, Lesson 9, students have opportunities to find products of rational numbers, irrational numbers, and a rational number and an irrational number. Guiding Questions found in Anchor Problems 1-3 of this lesson lead students to writing explanations and/or general rules for determining whether products will be rational or irrational. However, in Algebra 1, Unit 6, Lesson 10, students have opportunities to find sums of rational numbers, irrational numbers, and a rational number and an irrational number, but limited opportunities are provided for students to generalize a rule for whether the sums will be rational or irrational. 

  • A-SSE.3b: In Algebra 1, Unit 8, Lessons 2, 3, and 4 and Algebra 2, Unit 2, Lesson 6, students have opportunities to complete the square to write a quadratic function in vertex form, and students have opportunities to associate minimum and maximum function values with the vertex of a graph. Students have opportunities to identify the vertex of a graph using the vertex form of the function. However, opportunities are not provided for students to complete the square with the explicit intent of revealing the minimum or maximum value of the function.  

  • A-APR.6: In Algebra 2, Unit 4, Lesson 6, and Lessons 8-12, students have multiple opportunities to rewrite and simplify rational expression in different forms. However, there are limited opportunities for students to use long division to find quotients that have remainders which can be written in the form q+r(x)/b(x).

  • A-REI.7: In Algebra 1, Unit 8, Lessons 14 and 15, and Algebra 2, Unit 2, Lesson 11, teachers are advised to create their own problem sets for students. Although a list of types of problems to be included in the problem sets is provided, there are limited resources from which to create the problem sets.

  • F-BF.1b: In Algebra 1, Unit 3, Lesson 6, Target Task, students write the area of two triangles in a trapezoid and are instructed to write a formula for the area of the trapezoid using the area of the two triangles. In Algebra 2, Unit 3, Lesson 6, students are given practice with adding and subtracting polynomial functions. No opportunities were found for students to combine functions using multiplication and division.

  • S-ID.6a: In Algebra 1, Unit 2, Lessons 15-19, students have multiple opportunities to fit linear models to data represented by scatter plots. Students do not have sufficient practice with fitting non-linear (quadratic and/or exponential) function models to data. In Algebra 1, Unit 2, Lesson 15, the Problem Set contains one link (EngageNY Mathematics: Algebra 1, Module 2, Topic D, Lesson 13), that contains a discussion about quadratic and exponential function models, but not in the context of a data set. In Algebra 1, Unit 6, Lesson 18, there is an EngageNY Mathematics link (Algebra 1, Module 3, Topic B, Lesson 14, Example 3) that contains an opportunity for students to model data with an exponential function, but this lesson is not tagged with the standard. 

 Throughout the materials, there are some standards for which Guiding Questions and/or problems from the resources listed under Problem Set must be incorporated for the students to fully learn the standard. Examples include, but are not limited to:

  • N-CN.7: In Algebra 2, Unit 2, Lesson 7, students do not solve quadratic equations with real coefficients that have complex solutions, rather, they only identify such equations. In order for students to fully learn the standard, the EngageNY lesson linked in the Problem Set is needed. In addition, the Kuta free worksheets allow unlimited practice of solving quadratic equations with complex solutions.

  • F-IF.7a: In Algebra 1, Unit 7, Lesson 2, this standard is clearly addressed in the Criteria for Success, but students are not explicitly required to “show” the intercepts. The only situations where students are required to “show intercepts, maxima, and minima” are within the Problem Set Links.

  • S-ID.7: In Algebra 1, Unit 2, Lessons 17 and 19-22, students have limited opportunities to interpret slopes (rates of change) and the intercepts (constant terms) of linear models in data contexts. In order for students to have sufficient opportunities, the Guiding Questions and the problems from the extra resources are needed. For example, Lesson 17, the EngageNY lesson linked in the Problem Set is needed to interpret slope and y-intercept in context and is needed for students to fully learn the standard. 

Indicator 1c

Materials require students to engage in mathematics at a level of sophistication appropriate to high school.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The 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 of problems that allow students to engage in age appropriate contexts include:

  • Algebra 1, Unit 4, Lesson 4, Anchor Problem 1: Students analyze two graphs involving lemonade sales to determine which graph makes the most sense for predicting the number of sales needed to reach a fundraiser goal. Students interchange the independent and dependent variables to decide which graph is most sensible to use.

  • Algebra 2, Unit 6, Lesson 2, Anchor Problem 2: Students view a video referencing the Dan Meyer’s Ferris Wheel task. Students use graphs of trigonometric functions and their critical thinking skills to determine problem solutions.

  • Geometry, Unit 4, Lesson 19, Target Task: Students find the total length of a triathlon given that the race begins with a swim along the shore followed by a bike ride of specific length. After the bike ride, racers turn a specific degree and run a given distance back to the starting point. 

  • Geometry, Unit 8, Lesson 4, Anchor Problem 1: Students utilize a Venn diagram to display the coffee preferences of diner customers and calculate probabilities related to the preferences.  Students also determine characteristics of the events by answering a series of questions related to the probabilities (Which two events are complements of each other ?...etc.).

Examples of problems that allow students to engage in the use of various types of real numbers include:

  • Algebra 1, Unit 1, Lesson 2, Problem Set, Illustrative Math: “The Parking Lot”:  Students calculate parking lot charges at the rate of $0.50 per half hour over a varied number of times in minutes. They create a graph that represents the various parking costs (in decimal increments) for the parking times. Within the same lesson, Problem Set, Mathematics Vision Project: Secondary Mathematics One, Module 5: Systems of Equations and Inequalities, Lesson 5.4, students write, solve, and graph equations and inequalities involving use of fractions and decimals. 

  • Algebra 1, Unit 3: Linear Expressions & Single-Variable Equation/Inequalities, the Post Unit Assessment contains problems involving fractions that contain multiple variables with solutions that involve varied numbers, including decimals and fractions.

  • Algebra 2, Unit 5, Lesson 4: Students explore compound interest calculations to determine that the rate of growth approaches the irrational number defined as e.

  • Geometry, Unit 5, Lesson 1, Problem Set, EngageNY Mathematics Geometry, Module 4, Topic C, Lesson 10: “Perimeter and Area of Polygonal Regions in the Cartesian Plane”, students calculate area and perimeter of polygons graphed in the coordinate plane using various methods and varied numbers. For example, in Exercise 1, students find side lengths of a rectangle resulting in square root measures and use those measures to calculate the perimeter and area of the rectangle.

  • Geometry, Unit 7, Lesson 11, students practice finding arc lengths and must determine appropriate ways to use \pi in their calculations. Students make judgments as to what solutions are reasonably precise in relation to a given context.

Examples of problems that provide opportunities for students to apply key takeaways from Grades 6-8 include:

  • Algebra 1, Unit 3, Lesson 2, Target Task: Students are given two equations and are asked to know how they are equivalent without solving them (A-SSE.2 and A-REI.1). Equation A has all integer value coefficients and constants and Equation B has all rational number coefficients and constants (7.NS.1d and 7.NS.2c).

  • Algebra 1, Unit 4, Lesson 8, Target Task: Students write a linear inequality containing two variables to describe a context and graph the solutions in a coordinate plane (A-CED.3). This activity also builds on 7.EE.4.b, solving and graphing solutions for inequality word problems with one variable.

  • Algebra 2, Unit 5, Lesson 10, Target Task: Students calculate logs with the use of technology and must demonstrate their understanding of the values the calculator produces (F-LE.4). This activity connects with experiences students have had with 8.EE.4, where students interpret scientific notation generated by technology.

  • Algebra 2, Unit 8, Lesson 3, Anchor Problem 1: Students use tree diagrams to calculate conditional probabilities (S-CP.3). This activity builds on 7.SP.8b.

  • Geometry, Unit 4, Lesson 6, Target Task: Students demonstrate mastery of finding the sine of an angle and understanding that, by similarity in right triangles, the sine is consistent for a particular angle (G-SRT.6). In doing this, students are applying ratio reasoning (6.RP.3). 

  • Geometry, Unit 5, Lesson 12, Anchor Problem 1: Students work with dilations of polygons (triangle, parallelogram) and compare the areas of the original and scaled figures (G-GPE.7). This problem builds on 7.G.6.

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.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series.

Examples where the materials foster coherence within a single course include:

  • Algebra 1, Unit 3: Students work with algebraic expressions, linear equations and inequalities, many times in context and connect Algebra, Function, and Number standards. Specifically, in Lesson 7, students write equations in context (N-Q.1, A-CED.1 and A-CED.2). In the Target Task, students are given 4 linear equations and interpret the equations in order to answer specific questions about them (F-IF.5).

  • Algebra 2, Unit 3, Lesson 5, Anchor Problem 1: Students find the product of polynomials. (A-APR.1). In Anchor Problem 2, students identify both algebraically and graphically how fx multiplied by gx results in hx. The Guiding Questions make connections about the features of hx and how it relates to fx and gx (F-BF.3).

  • Geometry, Unit 3: Topic C in the Unit Summary describes how “students formalize the definition of ‘similarity,’ explaining that the use of dilations and rigid motions are often both necessary to prove similarity.” In Lessons 1-6, students first learn that similar triangles have proportional sides. Then, in Unit 4, Lesson 6, they learn through experimentation in Anchor Problem 1, that similar triangles all have the same sine ratio for the same angles. In Lesson 7, Anchor Problem 1, students make similar conclusions about cosine ratios. Finally, in Unit 4, Lesson 9, Anchor Problem 1, students determine relationships with respect to tangent ratios (G-SRT.6).

Examples where the materials foster coherence across courses include:

  • Algebra 1, Unit 3, Lesson 12, Anchor Problem 2: Students explain how to algebraically manipulate a compound inequality to solve a contextual inequality problem. In Geometry, Unit 1, Lesson 6, Anchor Problem 1, before solving equations with respect to geometric relationships, students review equation solving by explaining possible steps that could be used to solve the equation (A-REI.1). In Lesson 7, students use solving equations to solve for angle measures (G-CO.9). In Algebra 2, Unit 2, Lesson 5, Anchor Problem 1, students explain different ways they could find the roots of a quadratic function without using certain methods  (A-REI.1), and then find and describe the solutions (A-REI.4b).

  • Algebra 1, Unit 8, Lesson 1, Anchor Problem 1: Students learn to write quadratic functions in vertex form, first by identifying different written forms of the quadratic equation and describing the graph features that each form reveals. In Anchor Problem 2, students write a quadratic equation in vertex form using a graph. In Lesson 2, they transform a quadratic expression from standard form to vertex form (F-IF.4, F-IF.8). In Geometry, Unit 7, Lesson 3, students write the equation for a circle in standard form by completing the square (G-GPE.1). In Algebra 2, Unit 2, Lesson 6, students again complete the square, this time with an emphasis on seeing perfect square trinomials embedded in the vertex forms (A-REI.4).

  • Algebra 1, Unit 5, Lessons 12-16: The focus is on transformations of absolute value functions (F-BF.3). Later, in Algebra 1, Unit 8, Lessons 9 and 10, transformations with quadratic functions are found. Specifically, Lesson 10 focuses on transformations of quadratics in applications. In the Target Task, students are given a graph to model the scenario given and students must choose which representation(s) of transformations written using function notation will allow a ball to clear a wall (F-BF.3). In Geometry, Unit 1, Lessons 8, 9 and 10, transformations with rigid motion are found (G-CO.2, G-CO.4, G-CO.5, and G-CO.6). Specifically, Lesson 9 uses “algebraic rules to translate points and line segments.” Unit 2, Lesson 4 focuses on congruence of two dimensional polygons using rigid motion (G-CO.2, G-CO.4, G-CO.5, and G-CO.7). Unit 3, Lessons 1 and 2, focus on similarity and dilation which continues to connect transformations across the courses (G-SRT.2 and G-SRT.3). In Algebra 2,  Unit 4, Lesson 13, the focus is on transformations of rational functions (F-BF.3). Lastly, in Algebra 2, Unit 6, Lessons 9 - 11, the focus is on transformations of graphs of sine, cosine, and tangent functions (F-IF.7e, F-BF.3).

Indicator 1e

Materials explicitly identify and build on knowledge from Grades 6-8 to the High School Standards.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Throughout the curriculum Foundational Standards are cited which include standards from prior grades. In this series, the connections to Grades 6-8 standards can be found in the Guiding Questions. These standards are not simply being retaught, rather they are extended, thus providing those students who did not master the standard at grade level another opportunity to master that standard, and other students opportunities to recall and stretch what they learned previously.

Examples of lessons that allow students to build knowledge from Grades 6-8 to the High School Standards include:

  • Algebra 1, Unit 1, Lesson 1: 8.F.5 is cited as a Foundational Standard. In this lesson, students apply the idea of increasing and decreasing functions to analyze graphs of functions in contexts (F-IF.4). In addition to finding where the graphs are increasing or decreasing, students investigate average time per mile, the time it took for students to get to school, the distance the students live from the school, and more. 

  • Algebra 1, Unit 4, Lessons 11-13: Students build on 8.EE.8 by exploring methods of solving systems of equations (A-CED.3, A-REI.5, and A-REI.6). Specifically, in Lesson 12, students are given a method by which a system of equations was solved. The Guiding Questions allow students to explore various methods for solving systems to determine if the solutions will be the same. In Target Task 1, two systems of equations are given and students must determine if they result in the same solution without solving them. 

  • Algebra 2, Unit 8, Lesson 1: Four 7th-grade standards are cited as Foundational Standards.  Students build on their understanding of probability being a number between 0 and 1 (7.SP.5), approximating anticipated outcomes (7.SP.6), comparing results from an experiment to a probability model (7.SP.7), and finding probabilities of compound events using lists, tables, and trees (7.SP.8) by writing probabilities in P (desired outcome) notation and calculating probabilities for independent and mutually exclusive events (S-CP.1).

  • Geometry, Unit 2, Lesson 2: 8.G.5 is cited as a Foundational Standard. Students expand their informal arguments of angle sums and angles formed by parallel lines cut by a transversal to write formal proofs about triangles (G-CO.10).

  • Geometry, Unit 3, Lesson 9: Students build on 8.G.3. Students dilate figures where the center of dilation is not the origin and answer questions about their similarity (G-CO.2 and G-SRT.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.

Narrative Evidence Only
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA are not consistently identified, and are not consistently used to coherently support the mathematics which all students should study in order to be college and career ready.

The following standards are addressed in the materials, but are not explicitly identified as plus standards under the Lesson Map or Core Standards. Occasionally, in TIPS FOR TEACHERS, some of these standards are identified as plus standards. 

  • A-APR.7: In Algebra 2, Unit 4, Lessons 6 and 7 address A-APR.6 and A-APR.7. Both lessons focus on A-APR.6 with little to no problems addressing A-APR.7. While the connection is made with respect to working with fractions, no mention is made with respect to closure.

  • F-IF.7d: In Algebra 2, Unit 4, Lessons 8-13, students identify asymptotes and end behavior.

  • F-BF.4c: In Algebra 2, Unit 2, Lesson 4, students read values of inverse functions from graphs and tables.

  • F-BF.5: In Algebra 2, Unit 5, Lesson 9, students explore the relationships between exponential functions and logarithms.  

  • F-TF.3: In Algebra 2, Unit 6, Lesson 7, students use special triangles to determine geometrically the values of sine, cosine, tangent for \pi/3, \pi/4 and \pi/6. This lesson also addresses F-TF.1 and is embedded in the material with the plus standards. In addition, this is the only lesson in which this standard is included, so it is not a lesson that can be skipped. In lesson 8, students continue this work with \pi-x, \pi+x, and 2\pi-x in terms of their values for x, where x is any real number. 

  • F-TF.4: In Algebra 2, Unit 6, Lesson 9, students consider whether sine and cosine functions are even or odd.

  • F-TF.6: In Algebra 2, Unit 7, Lesson 4, students are given several functions that are equivalent. 

In the guiding questions, students determine how the domain could be restricted so that the inverse could only have one solution. This lesson also addresses F-TF.7.

  • F-TF.7: In Algebra 2, Unit 7, Lessons 5 and 6, students use a graphing calculator to evaluate solutions and interpret the terms. 

  • F-TF.9: In Algebra 2, Unit 7, Lessons 11-14, F-TF.9 is listed as a Core Standard. Lessons 11-12 contain problems involving addition and subtraction formulas for sine, cosine and tangent. In Lesson 13, students use the sum formulas to derive the double angle formulas. Lesson 14 also lists F-TF.9 as a Core Standard, but the lesson does not address the standard.

  • G-SRT.9: In Geometry, Unit 4, Lesson 16, students need to derive the area formula for any triangle in terms of sine. Through the Anchor Problems and the Guiding Questions, students also find the area of a triangle, given two sides and the angle in between. Through the Guiding Questions, students derive the formula for the area of a triangle, A= \frac{1}{2}ab sinC. 

  • G-SRT.10: In Geometry, Unit 4, Lessons 17 and 18, students use the Law of Sines and the Law of Cosines to solve side lengths and/or angles of triangles. In Lesson 17, the Guiding Questions allow for students to verify the Law of Sines algebraically. In Lesson 18, the Notes of Anchor Problem 1, states, “Students should algebraically verify the Law of Cosines during this Anchor Problem and class discussion.” In Algebra 2, Unit 7, Lessons 15 and 16, students use the Law of Sines and Law of Cosines to find angle and side measures of acute triangles. 

  • G-SRT.11: In Geometry, Unit 4, Lessons 16-19, students solve real-world problems using the Law of Sines and the Law of Cosines.

  • G-C.4: In Geometry, Unit 7, Lesson 9, G-C.2 is also addressed. In the Tips for Teachers, states that this standard, “ is represented in its fullest in the problem set guidance.” This lesson can be taught without doing the Problem Set(s) that address the plus standard.

  • G-GMD.2: In Geometry, Unit 6, Lessons 10-12: Lesson 10 addresses G-GMD.1 and G-GMD.2. This lesson focuses on Cavalieri’s Principle and uses it to derive the formula of the volume of a sphere. Lesson 11 addresses G-GMD.2 and G-GMD.3 and G-GMD.2 is identified as a plus standard in the Tips for Teachers. The lesson investigates how slices and/or cross sections of pyramids are related to pyramids. In Anchor Problem 2 of this lesson, a rectangular pyramid is given and students find the area of the base and the area of the cross section and then look at the relationship between the volume of the top part of the pyramid, above the cross section and the full pyramid. G-GMD.2 is integrated throughout Lesson 12 which addresses G-GMD.1, G-GMD.2, G-GMD.3, and N-Q.3. In Anchor Problem 2, there is an extension that would directly relate to G-GMD.2. 

  • S-CP.8: In Geometry, Unit 8, Lesson 3, students apply the general Multiplication Rule, P(A and B)=P(A)P(B|A)=P(B)P(A|B), and use it to solve context problems.

  • S-CP.9: In Geometry, Unit 8, Lessons 9 and 10, combinations and permutations are covered. 

Additionally, at the beginning of some units, plus standards may be listed on the Unit Summary page under Future Standards, but are not found in the materials. For example, A-APR.5 is listed under Future Standards in Algebra 1, Unit 7 Summary, but is not found in the materials. 

Plus standards not mentioned in this report were not found in the materials.

Gateway Two

Rigor & Mathematical Practices

Meets Expectations

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Gateway Two Details

The materials reviewed for Fishtank Math AGA meet expectations for rigor and balance. The materials meet expectations for providing students opportunities in developing conceptual understanding, procedural skills, and application, and the materials also meet expectations for balancing the three aspects of rigor. The materials meet expectations for Practice-Content Connections as the materials intentionally develop all of the mathematical practices to their full intent. However, the materials do not explicitly identify the mathematical practices in the context of individual lessons, so one point is deducted from the score in indicator 2e to reflect the lack of identification.

Criterion 2a - 2d

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

8/8
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for rigor and balance. The materials meet expectations for providing students opportunities in developing conceptual understanding, procedural skills, and application, and the materials also meet expectations for balancing the three aspects of rigor.

Indicator 2a

Materials support the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for developing conceptual understanding of key mathematical concepts especially where called for in specific content standards or cluster headings. Lessons in the materials have Anchor Problems that include Guiding Questions for teachers to ask. These questions often assist with developing conceptual understanding. The Guiding Questions also help students find solutions and write explanations to support those solutions.

Examples of lessons that provide students with opportunities to demonstrate conceptual understanding include:

  • Algebra 1, Unit 4, Lesson 7: In Anchor Problem 1, students write a linear inequality for a graph. Guiding Questions elicit students’ justifications for their choices of inequalities (A-CED.3). Next, the questions prompt students to consider variations that would occur in their inequality with specific changes to the graph. Finally, students consider how restrictions in the domain and range would alter the solutions shown in the graph.

  • Algebra 1, Unit 5, Lesson 12: Students investigate transformations of functions (F-BF.3). Anchor Problems 2 and 3 were adapted from a Desmos activity, Introduction to Transformations of Functions. In the Anchor Problems, students are instructed to go to the slides that correspond to the problems. Specifically, Anchor Problem 3 has a link to slide 7 on the Desmos activity. On the slide, students have a slider that changes the value of k in the function f(x)=|x|+k. Students answer three questions on the slide asking what happens when k is zero, negative, and what effect k has on the function. Also, the Problem Set includes a link to a Desmos activity: Absolute Value Translations, where students investigate vertical translations with absolute value functions. In the activity, students begin by graphing the function, f(x) = x and compare it with f(x) = x. Subsequent slides have students predict what functions will look like given k values for vertical/horizontal translations and then graph the function on the next slide to check their thinking.

  • Algebra 2, Unit 3, Lesson 10: In Anchor Problem 1, students develop conceptual understanding through a series of images of diagrams of cubes with a portion cut out to show the total volume representing A^3 -B^3(A-APR.4). Then the lesson continues to further develop students’ understanding of the proof of A^3 -B^3=(A-B)(A^2+AB+B^2) with the use of Guiding Questions and visuals of the cubes. 

  • Algebra 2, Unit 8, Lesson 2: In Anchor Problem 1, students use a Venn diagram to represent coffee preferences of diner customers. The diagram and Guiding Questions are used to develop students' conceptual understanding about determining whether two events are mutually exclusive or not, and assist students with developing conceptual understanding of unions, intersections, and complements of events (“or,” “and,” “not”) (S-CP.1).

In Anchor Problem 2, other types of diagrams are used to help students further develop a conceptual understanding of mutually exclusive and non-mutually exclusive events, and the 

probability addition rules that relate to these types of events. When two events are mutually exclusive, P(A or B) =P(A) + P(B) When two events are not mutually exclusive, P(A or B) = P(A) +P (B)-P(A and B) (S-CP.7).

  • Geometry, Unit 2, Lesson 9: Students investigate triangle congruence using rigid motions (G.CO.7 and G.CO.8). In Anchor Problem 1, students use patty paper to trace two sides and an included angle. Then they explore how many different triangles can be made, starting with those two sides and the included angle. Guiding Questions prompt students to discover that all the triangles they make are congruent. In Anchor Problem 2, students answer “How can you use rigid motions to prove that if two triangles meet the side-angle-side criteria, the triangles are congruent?” The Guiding Questions, when used, support conceptual understanding by asking students “what properties of rigid motions show that the corresponding line segments and corresponding angles are congruent?” thus connecting rigid motions with congruent triangles. 

  • Geometry, Unit 5, Lesson 5: The Problem Set contains a link to Open Middle - Parallel Lines and Perpendicular Transversals: Students use the digits 1-9 (at most once each), to fill in open boxes to complete three equations. The digits become coefficients of the x and y terms in the equations.Two equations should represent parallel lines and the third equation should represent a transversal that is as close to being perpendicular to the parallel lines as possible. (G-GPE.5).

Indicator 2b

Materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters.

Throughout the materials, there are Problem Sets which link to a variety of resources. Teachers can select problems and activities that align with the lessons with the use of these resources.  Most of the opportunities for students to develop procedural skills related to the lessons are found in the linked resources. Students have limited practice with solving problems using the Anchor Problems and the Target Tasks provided in each lesson. However, for some lessons, the resource links are limited. Teachers are given instructions for what types of problems to include in an assignment and create their own sets of problems and activities.

Examples of lessons that provide opportunities for students to develop procedural skills include:

  • Algebra 1, Unit 1, Lesson 5: The Anchor Problems for this lesson give students opportunities to calculate average rates of change for functions through the use of multiple representations (graph, table, and context problem) (F-IF.6). Students are also asked to determine over which intervals functions are increasing or decreasing and whether or not the functions are linear.

  • Algebra 1, Unit 7, Lessons 6 and 7: Students develop procedural skills for factoring quadratic equations. In Lesson 6, students first practice factoring quadratic expressions with a leading coefficient of one, and explore the relationship between the factors of the constant term and the coefficient of the middle term. Then students use their factoring skills to solve quadratic equations. In Lesson 7, students practice factoring quadratic expressions with leading coefficients that do not equal one and then use those skills to solve quadratic equations. Problem Sets for both lessons contain links to Engage NY lessons, Kuta Worksheets, and other resources that provide opportunities for students to practice factoring quadratics. Additionally, the Target Tasks in both lessons provide opportunities for students to show mastery of factoring different trinomials, including trinomials that have a greatest common factor, as well as one that has a leading negative coefficient (A-SSE.1a and A-SSE.3a). 

  • Algebra 2, Unit 3, Lesson 5: Students have opportunities to develop their procedural fluency with finding products of polynomials and representing the products graphically (A-APR.1). through the Problem Set. There are links for students to practice the procedural skills of polynomial operations in Engage NY Lesson, Mathematics Vision Project Modules and Kuta Worksheets. Through Guiding Questions, students are able to link the degrees of polynomials and polynomial factors to key graph features.

  • Geometry, Unit 1, Lesson 3: Students develop procedural skills for constructing angle bisectors. The Problem Set links provide students with additional opportunities to practice with this geometric construction with the use of Engage NY Lesson links (G-CO.12).

  • Geometry, Unit 2, Lesson 13: The Problem Set has links to two Engage NY lessons that have a note on both to “ask students to describe the transformations that will map the “givens” and show congruence”. This provides students opportunities to develop procedural skills with using congruence and similarity to prove relationships in geometric figures (G.SRT.5). 

  • Geometry, Unit 7, Lesson 12: Students use the proportional relationships between the radius of a circle and the length of an arc in preparation for converting degrees to radians. Both the Practice Set and Target Task provide students with opportunities to practice this skill. Later, in Algebra 2, Unit 6, Lesson 7, students again learn to convert between degrees and radians and have many opportunities to practice this skill using the links provided in the Problem Set and the Target task (G-C.5).

Indicator 2c

Materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Examples of lessons that include multiple opportunities for students to engage in routine and non-routine mathematics applications include:

  • Algebra 1, Unit 1, Lesson 9: The Problem Set includes Yummy Math, “Harlem Shake” problem, in which students are presented with a situation in which a group of students want to investigate the lifespan of an internet meme using the song, “The Harlem Shake.” Students are given a graph of data and are asked a series of questions about the graph. Students find how many days it took for the video to be mentioned 100,000 and then how much longer it took to get 200,000 mentions. Students must also determine if the data shows linear growth, the greatest rates of growth, and more. Lastly, students determine how many total mentions the post received (A-CED.2, F-IF.5, F-LE.3). 

  • Algebra 1, Unit 5, Lesson 5: For Anchor Problem 2, students are given a context problem involving bank account transactions. Students analyze a graph that is drawn to represent the transactions. However, this graph is misleading and inaccurate, because it represents the transactions as a continuous function. Based on the given information and a series of Guiding Questions, students determine the inaccuracies in the graph, and are prompted to draw a new graph in the form of a step-function that more accurately represents the problem context. Students are also asked to represent the context of the problem algebraically. This problem addresses graphing and writing step-functions contextually (A-CED.3, F-IF.7b).

  • Algebra 2, Unit 5, Lesson 2: In Anchor Problem 2, students are given the situation where a fisherman introduces fish illegally into a lake, and the growth of the species is modeled by an exponential function. Students need to use the function to calculate how many fish were released initially; given the number of fish present after a specific time, find the base of the function; and if the base is known, calculate the weekly percent growth rate and interpret what this means in everyday language (F-IF.8b). 

  • Geometry, Unit 6, Lesson 17: The Problem Set links students to the Illustrative Mathematics problem: “How many cells are in the Human body?” “The purpose of this task is for students to apply the concepts of mass, volume, and density in a real-world context.” The given information includes facts about the volume and density of a cell which are used to compute the mass of the cell. Students need to work with mass, density and volume in real life context and understand what information they need to know and what information they can make assumptions about. Students will need to understand what are reasonable assumptions to be made with this problem and they will need to convert the weight of a person into grams (N-Q.2, N-Q.3, G-GMD.3, G-MG.2).

  • Geometry, Unit 8, Lesson 8: In the Anchor Problem, students make decisions about medical testing based on conditional probabilities. Students must complete a two-way frequency table and a tree diagram to make decisions about medical test results for a certain population percentage. Students need to find the number of people in a random sample of 2,000 people who take the test for the disease who get results that are true positive, false positive, true negative, and false negative. Students must also determine: which of the given information is conditional and which is not conditional; probabilities of getting certain test results; whether certain test results classify as independent or dependent events. Guiding Questions include: What are the risks and rewards for taking this test?; Are testing positive and having the disease independent or dependent events?; and Are testing negative and not having the disease independent or dependent events? (S-CP.4 and S-CP.5)

Indicator 2d

The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

2/2
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Indicator Rating Details

he materials reviewed for Fishtank Math AGA meet expectations for including lessons in which the three aspects of rigor are not always treated together and are not always treated separately. The aspects of rigor are balanced with respect to the standards being addressed. Examples of lessons that engage students in the aspects of rigor include:

  • Algebra 1, Unit 1, Lesson 6: The Problem Set provides a link to MARS Formative Assessment Lessons for High School Representing Functions of Everyday Situations which focuses on conceptual development of representing context situations as functions and graphs (F-IF.4). In the task, students work towards understanding how specific situations will look as a function on a graph. Students work together in collaborative groups to match situations to graphs. After this is completed, students then will match functions to their paired graphs/situations. Throughout the lesson, there are questions for the teacher to ask students in order to further their understanding.

  • Algebra 1, Unit 7, Lesson 10: The Anchor Problems assist students with developing their conceptual understanding of solutions to quadratic equations by having students identify the roots to a quadratic equation with the use of three representations (equation, graph, table) (A-SSE.3a, F-IF.9). The Problem Set provides a link to Illustrative Mathematics, where students work on proving the zero product property. Additionally, links to Kuta worksheets are provided to give students practice with solving quadratic equations. In Algebra 1, Unit 7, Lesson 13, students have opportunities to interpret quadratic solutions in context problems (example: Using a quadratic equation and its graph to model the height of a ball as a function of time) (A-SSE.3a,F-IF.8a).

  • Algebra 2, Unit 4, Lesson 16: In this lesson, Anchor Problems 1 and 2 have students explore two methods of solving rational equations; clearing the equation of fractions by multiplying each term in the equation by the least common denominator and solving the resulting equation, or by rewriting each term in the equation with a common denominator and setting the numerators equal. Students determine if the resulting solution is valid or extraneous (A-REI.2). Then students determine how each of these types of solutions would be interpreted graphically. Students answer: If the solution is valid, how can the y-coordinate be determined? If the solution is extraneous, what does this mean graphically? The Problem Set provides links to lessons that contain additional problems with which students can develop their procedural skills for solving rational equations and identifying the types of solutions.

  • Algebra 2, Unit 5, Lesson 3: Anchor Problem 1 gives a real-world scenario that is modeled by an exponential function. Students write the equation, which is a procedural skill within the application. The Guiding Questions asks questions to check procedural skill and conceptual understanding. Students find the height after a given time and are asked how they know their solutions are correct. Anchor Problem 2 has two bank options for students to choose which would be better after ten years with a given investment and percentage rate (F-BF.1a, F.LE.5). 

  • Geometry, Unit 4, Lesson 12: Students use the Pythagorean Theorem and right triangle trigonometry to solve application problems (G-SRT.8). The Anchor Problems and the Target Task provide opportunities for students to solve application problems that focus on solving right triangles. For example, the Target Task has students find how far a friend will walk to meet a friend who is ziplining down from a building. The Problem Set provides suggestions and links to lessons that also focus on application problems involving right triangles. 

  • Geometry, Unit 5, Lesson 12: In Anchor Problem 1, students develop a conceptual understanding of G-GPE.7 through the Guiding Questions. Students use dilations to create similar figures of original polygons using given scale factors (G-SRT1b). Students compare areas of original and scaled figures to determine the relationship between the areas of the figures and the scale factors. Then students develop a general rule that represents this relationship (The value of the ratio of the area of the scaled figure to the area of the original figure is the square of the scale factor of dilation). The Problem Set provides links to EngageNY and CK12 resources that include additional practice problems for students to practice their procedural skills.

Criterion 2e - 2h

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

7/8
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for Practice-Content Connections as the materials intentionally develop all of the mathematical practices to their full intent. However, the materials do not explicitly identify the mathematical practices in the context of individual lessons, so one point is deducted from the score in indicator 2e to reflect the lack of identification.

Indicator 2e

Materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

1/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA partially meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the Standards for Mathematical Practice. The Standards of Mathematical Practice are listed at the end of each Unit Summary, along with the Common Core Standards and Foundational Standards identified for the particular unit. Guiding Questions found within each lesson reflect the use of the Standards of Mathematical Practice for assisting students with solving problems, but they are not explicitly identified in the context of the individual lessons for teachers or students. As a result of this one point is deducted from the scoring of this indicator.

Examples of where and how the materials use MPs 1 and/or 6 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

  • Algebra 1, Unit 6, Lesson 19: Students write exponential growth functions to model financial applications that include compound interest. Guiding Questions assist students with applying the correct formulas to determine account interests and balances to solve the Anchor Problems and Target Task found in the lesson. Students must persevere to solve the problems by identifying the meaning of the variables found in the formulas and assigning the correct values to them, and attend to precision when expressing the solutions to the problems (MPs 1 and 6); (F-IF.8, F-LE.2, F-LE.5).

  • Algebra 1, Unit 8, Lesson 15: For Anchor Problem 3, students work with a word problem where they need to write two equations - one for a ball that is thrown, and one for a remote controlled toy plane that takes off at a constant rate. Students need to think through how those two functions are similar and how they are different. They will need to consider what type of function each of these will represent (MP1). The Guiding Questions walk students through some questions that will help students think through some of the process - asking students - what type of function represents the height of the ball, linear, exponential, or quadratic? Additionally the question asks what graph shape represents the motion path of the toy plane? Students must make a prediction as to whether the plane and the ball will ever reach the same height, explain the reasoning for their prediction, and sketch a graph to represent their prediction. Students must attend to precision by correctly using the units found in the problem context (MP6); (A-REI.7, A-REI.11).

  • Algebra 2, Unit 6, Lesson 14: The Anchor Problems link to a Desmos activity called “Burning Daylight.” In this activity, students need to work through the problems to find the period, amplitude, midline and phase shift through the context of geography and how much sunlight different areas of the country have (F-TF.3, F-TF.5). Students need to make sense of the problem they are given and understand the context of the graphs they are given to answer the questions in the Desmos problems, then write equations for the hours of sunlight for the given locations (MP1). Additionally, after students make their equation and graph related to one city in Alaska, the context of the problem changes to a more northern city in Alaska, which allows students to write another function, then see the actual graph and make sense of this graph. Students persevere to solve this problem by accurately interpreting the graphs associated with the problem and attending to precision when using the formulas needed to process problem solutions (MP1 and MP6). 

  • Algebra 2, Unit 8, Lesson 10: Students describe and compare statistical study methods. For each Anchor Problem in this lesson, students need to use the problem description to determine what type of study (survey, observational or experimental), will result in the most reliable data (S-IC.3). Students must also identify population parameters, and determine advantages and disadvantages of using one type of study compared to another (MP1). In Anchor Problem 1 students must determine if a correlation exists between the number of minutes a train is delayed and the number of violent acts that occur on the platform or on the train. Students are given various scenarios of how to conduct this study and must identify what type of study each scenario represents and which would result in the most reliable data. Guiding Questions assist students with analyzing the problem to determine which scenario would result in the most reliable data (MP6).

  • Geometry, Unit 4, Lesson 2: The Problem Set provides a link to an Illustrative Mathematics task that contains 3 right triangles surrounding a shaded triangle. Students need to prove that the shaded triangle is a right triangle. Students must determine an approach for solving the problem. Throughout this proof, students need to both attend to precision (by keeping their answers in irrational numbers, not rounding them) and persevering through the calculations to complete the proof (MP1 and MP6). Students must also attend to precision when using the Pythagorean Theorem to find the hypotenuse of the smaller triangle to find some of the missing information needed to solve the problem (G-SRT.4). 

  • Geometry, Unit 5, Lesson 10: In this lesson, students find areas and perimeter of polygons sketched in the coordinate plane. Students must also identify and write the correct formulas needed to find the areas and perimeters of the polygons. Guiding Questions assist students with persevering to plan strategies for solving the problems and attending to precision when using the coordinates of polygon vertices to find the measures needed to find the areas and perimeters of the polygons (MP1 and MP6); (G-GPE.7).

Indicator 2f

Materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the Standards for Mathematical Practice.

Examples of where and how the materials use MPs 2 and/or 3 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

  • Algebra 1, Unit 4, Lesson 6: In Anchor Problem 2, students reason abstractly and quantitatively as they analyze the graph of an inequality to determine solution points. Students are told to name a point that is part of the solution, then explain their reasoning using the graph, and the algebraic expression for the inequality. Additionally, they need to do the same with a point that is not a solution, explaining their reasoning both graphically and algebraically (MP2). Guiding Questions assist students in constructing arguments to justify their conclusion. (MP3); (A-REI.12).

  • Algebra 1, Unit 4, Lesson 9: The Problem Set contains a link to Illustrative Mathematics: Estimating a Solution via Graph. In this lesson, students reason quantitatively and graphically to analyze a given solution to a system of equations to determine why the given solution is incorrect. Calculating the slopes and the y-intercepts for each of the equations indicates there is an unique solution to the system of equations. Using the graph of the system of equations shows that the intersection point (solution) for the system of equations is to the right of the y-axis and below the x-axis, indicating a positive x-coordinate and a negative y-coordinate.The given solution has a positive x-coordinate much less than the one indicated in the graph, and a positive y-coordinate (MP2 and MP3); (A-REI.6 and A-REI.11).

  • Algebra 1, Unit 6, Lesson 16: In the Problem Set there is a link to Illustrative Mathematics: Boiling Water. In this problem, students are challenged to compare two related data sets that are modeled with a linear function, but when the data sets are combined, the combination is better represented by an exponential function (F-LE.1 and F-LE.2). Within the exploration, students consider why the data seems inaccurate. Then students construct an improved estimate for the slope of a linear equation to fit the data through the process of contextualizing and decontextualizing information in order to generalize a pattern that is not immediately clear (MP 2 and MP3).

  • Algebra 2, Unit 2, Lesson 4: In Anchor Problem 2, students are presented with the information about how a quadratic equation is increasing and decreasing over specific intervals and that it has a rate of change of zero at x=3. Students need to then describe how each of three given graphs fit the description. The Guiding Questions ask students what is the same about each graph and what is different. Students then reason through how the different graphs can all look different yet meet the same criteria given in the Anchor Problem (MP2); (F-IF.4, F-BF.3).

  • Algebra 2, Unit 5, Lesson 2: In Anchor Problem 1, students are given a graph and an equation that represents the graph in the form of f(x)=ab^x  and asked to find the values of a and b.

Students must use the graph to test the accuracy of their equation, identify features of the exponential function they see in the graph, determine rates of percent change between points on the graph, and whether the graph represents exponential growth or decay (MP2); (F-IF.4 and F-LE.2). 

  • Geometry, Unit 3, Lesson 5: The Problem Set contains a link to an Illustrative Mathematics lesson, Dilating a Line, in which students need to locate images based on the dilation they perform, and interpret what they think will happen to the line when they perform the dilation. In this task, they are verifying their work experimentally, then verifying their work with a proof. This helps students to construct their arguments and explain their reasoning to better understand the mathematical operation they are performing (MPs 2 and 3); (G-SRT.1).  

  • Geometry, Unit 3, Lesson 18: In this lesson,Tips for Teachers contains a link to Shadow, A Solo Dance Performance Illuminated by Three Synchronized Spotlight Drones. An opening modeling scenario is provided in which students consider the behavior of three different spotlights directed onto a ballet performance (G.SRT.5). Students may reflect on a previous lesson, Deducting Relationships: Floodlight Shadows, where spotlights were placed at various angles and consider how the spotlight arrangements affect the appearance of the performance shown in a video.The video can be paused and restarted to allow students to assess the reasonableness of their calculations. Reflection on prior learning, guides students to manipulate symbolic representations to explain the visual effects (MP 2).

Indicator 2g

Materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the Standards for Mathematical Practice. 

Examples of where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

  • Algebra 1, Unit 2, Lesson 5: The Target Task has four questions that begin by having students draw a dot plot “representing the ages of twenty people for which the median and the mean would be approximately the same.” Students are not given a data set or any other information about what numbers should be used. Students need to make some assumptions in order to come up with their dot plot that meets the criteria. Question 2 asks the same thing, but students need to find a dot plot where “the median is noticeably less than the mean”. Students then have to determine and explain “which measure of spread” would be used for each data set. Lastly, students have to calculate the variance in the data sets they created in Questions 1 and 2 (MP 4); (S-ID.3). Students are free to choose the tool(s) that they feel would be appropriate when creating the dot plots and calculating the variance (MP 5).

  • Algebra 1, Unit 6, Lesson 17: The Problem Set contains a link to Engage NY, Algebra 1, Module 3, Topic A, Lesson 5 - Problem Set. Problem #6 of the Problem Set involves two band members who each have a method for spreading the word about their upcoming concert. Students have to show why Meg’s strategy will reach less people than Jack’s in part a. In part b, students explore if Meg’s strategy will ever inform more people than Jack’s if they were given more days to advertise. Lastly, in part c, students revise Meg’s plan to reach more people than Jack within the 7 days (MP 4); (F-LE.1). 

  • In Algebra 2, Unit 2, Lesson 3: In the lesson, students are factoring quadratics to find the roots and other features of a quadratic. (F-IF.8 and A-SSE.3). One of the Criteria for Success states, “Check solutions to problems using a graphing calculator.” Graphing calculator use is also incorporated into the guiding questions in Anchor Problem 1 (MP 5).

  • Algebra 2, Unit 7, Lesson 8: In the Target Task, students are given estimated populations of rabbits and coyotes, as well as the graphs of the data. The students need to write a function for each of the simulations and then calculate two different numbers of years where the population estimate will reach specific quantities for the rabbits and the coyotes (MP4); (F-TF.7).

  • Geometry, Unit 1, Lesson 3: In Anchor Problem 2 students first place a set of construction instructions into the correct order. Then students follow the instructions to copy and construct an angle using a compass and straightedge (MP5); (G-CO.12).

  • Geometry, Unit 6, Lesson 7: In Anchor Problem 2, the dimensions of a cylindrical glass filled with lemonade are given. Students need to determine how many cone-shaped cups, half the height of the glass and the same radius can be completely filled with the lemonade from the glass (G-GMD.3). Guiding Questions help students estimate and form conjectures about how to solve the problem (MP 4), and select appropriate formulas to solve the problem (MP 5).

Indicator 2h

Materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the Standards for Mathematical Practice.

Examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • Algebra 1, Unit 5, Lesson 7: Students construct understanding on solving absolute value equations by looking at graphs. In Anchor Problem 1, students consider the equation, |x|=5 and look at it as a system of equations where f(x)=|x|, g(x)=5, and f(x)=g(x). In the Guiding Questions, students are asked to consider what each graph looks like and where the two functions intersect. Lastly, students are asked what the solution to |x|=5 looks like on a number line (A-REI.1 and A-REI.11).

  • Algebra 2, Unit 7, Lesson 1: In Tips for Teachers, there is a link to Sam Shah’s blog post: Dan Meyer Says Jump and I Shout How High? In his post, Sam describes having students graph functions that seem to be different but then produce the same graph. Then Sam provides triangles from which students produce symbolic logic that represents the equality in the graphs. Finally, students produce the algebraic proof for the logic (F-TF.8).

  • Geometry, Unit 3, Lesson 6: In Anchor Problem 1, students explore a diagram containing two triangles drawn on a sheet of lined paper. The two triangles are contained in one figure and the bases of the triangles coincide with the lines on the paper, which represent parallel lines. Students are asked what they would need to know to justify that the triangles are dilations of one another. Students attempt to answer the question with the use of the dilation theorem and the side-splitter theorem, by explaining how each of these theorems applies to the diagram (G-SRT.5).

Examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

  • Algebra 1, Unit 6, Lesson 6: Students engage with N-RN.1 through a series of questions. Students consider why the equation 100^\frac{1}{2}=50 is not true.To help answer this question, students are given a pattern of 100 raised to different integer exponents, and asked where \frac{1}{2} fits into this pattern? Then students are asked to rewrite the base 100 as a power of 10 and place it in the equation (10^2)^\frac{1}{2}=? to see if students recognize what exponent rule can be applied and what the outcome would be. Students are then asked to try evaluating other expressions that contain rational exponents. Through the Guiding Questions, students are further asked to explore their conceptual understanding by asking what they think it means to have a fractional exponent, specifically, what does the denominator of the fractional exponent indicate?

  • Algebra 2, Unit 6, Lesson 8: In Anchor Problem 1, students rewrite angle values, such as in sin\frac{11\pi}{6} ,in the form of 2\pi-x  to find their values. As they do so, they relate their revisions to angles on the unit circle. They generalize this relationship and process to solving problems in the problem set link to Engage NY Mathematics: Precalculus and Advanced Topics, Module 4, Topic A, Lesson 1 (F-TF.3).

  • Geometry, Unit 4, Lesson 9: In Anchor Problem 2, students are asked to describe why the tangent of 90° is undefined. With the use of special right triangles, students explore what the tangent of the following angles: 0°, 45°, 60° and 90° would be. Students look at the pattern of these, then describe what happens to the tangent as the value of the angle approaches 90° and as it approaches 0°.To extend the understanding of why the tangent of 90° is undefined, students are then asked what is the relationship between tangents of complementary angles (G-SRT.6, G-SRT.7).

Gateway Three

Usability

Does Not Meet Expectations

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Gateway Three Details

The materials reviewed for Fishtank Math AGA do not meet expectations for Usability. The materials partially meet expectations for Criterion 1, Teacher Supports, do not meet expectations for Criterion 2, Assessment, and do not meet expectations for Criterion 3, Student Supports.

Criterion 3a - 3h

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

7/9
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA partially meet expectations for Teacher Supports. The materials: include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities. The materials partially contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject, and they partially include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives.

Indicator 3a

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

1/2
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Indicator Rating Details

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.

Indicator 3b

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

1/2
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Indicator Rating Details

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.

Indicator 3c

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. Generalized correlation information is present for the mathematics standards addressed throughout the series and can be found in the course summary standards map, unit summary lesson map, and the list of standards identified for each lesson. Examples include:

  • In Algebra 1, Algebra 2, and Geometry, a Standards map for each course includes a table with each course-level unit in columns and aligned standards in the rows. Teachers can easily identify a unit and when each standard will be addressed. 

  • In most lessons, there is a list of content standards following the lesson objective. For example, in Algebra 1, Unit 1, Lesson 2, the Core Standards are identified as F.IF.1 and F.IF.2. The Foundational Standards are identified as 8.F.1. 

  • Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Problems, Problem Set, and Target Task. Occasionally these contain additional references to standards. For example, in Geometry Unit 5, Lesson 5, the Tips for Teachers connects 4.G.2 “Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines” with the high school geometry lesson objective of “Describe and apply the slope criteria for parallel lines”.

Each Unit Summary includes a narrative outlining relevant prior and future content connections for teachers. There is also a Lesson Map which gives an objective and the standard(s) for each lesson. Examples include:

  • The Algebra 1, Unit 2 summary includes an overview of how the unit builds from prior coursework. The materials state, “Scatterplots are explored heavily in this unit, and students use what they know about association from 8th grade to connect to correlation in Algebra 1.”

  • The Geometry, Unit 8 summary includes an overview of how the content learned will form a foundation for future learning. The materials state, “In Algebra 2, students will continue their study of probability by studying statistical inference and making decisions using probability.”

  • The Algebra 2, Unit 7 summary includes an overview that indicates trigonometry is needed for Calculus, yet also states that the unit builds on the previous unit on trigonometry functions to expand students’ knowledge.

Indicator 3d

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Narrative Evidence Only
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials do not contain strategies for informing students, parents, or caregivers about the mathematics their student is learning. Additionally, no forms of communication with parents and caregivers and no suggestions for how stakeholders can help support student progress and achievement were found in the materials.

Indicator 3e

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

2/2
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Materials explain the instructional approaches of the program, or materials include or reference research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found within Our Approach and Math Teacher Tools. Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

  • Under the About Us section, there is a link to Our Approach, which includes a reference to “best practices,” the Common Core State Standards and Massachusetts Curriculum Frameworks. The approach is stated as being one of flexibility for teachers to be able to adapt lessons. Well-known open educational resources are mentioned as being included in the Fishtank materials.

  • Within Math Teacher Tools, there is a section called, Preparing To Teach Fishtank Math, the  Understanding the Components of a Fishtank Math Lesson section, outlines the purpose for each lesson component. It states that, “Each Fishtank math lesson consists of seven key components, such as the Objective, Standards, Criteria for Success, Tips for Teachers, Anchor Tasks/Problems, Problem Set, the Target Task, among others. Some of these connect directly to the content of the lesson, while others, such as Tips for Teachers, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” 

  • In Math Teacher Tools, Assessments, Overview, Works Cited lists, “Wiliam, Dylan. 2011. Embedded formative assessment. Bloomington, Indiana: Solution Tree Press.” and “Principles to Action: Ensuring Mathematical Success for All. (2013). National Council of Teachers of Mathematics, p. 98.”

  • In Teacher Tools, there is a link to Academic Discourse. The Overview outlines the role discourse plays within Fishtank Math. The materials state, “Academic discourse is a key component of our mathematics curriculum. Academic discourse refers to any discussion or dialogue about an academic subject matter. During effective academic discourse, students are engaging in high-quality, productive, and authentic conversations with each other (not just the teacher) in order to build or clarify understanding of a topic.” 

Additional documents, “Preparing for Academic Discourse,” “Tiers of Academic Discourse,” and “Strategies to Support Academic Discourse,” that may provide more detail are not available in Fishtank Math AGA.

Indicator 3f

Materials provide a comprehensive list of supplies needed to support instructional activities.

1/1
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for providing a comprehensive list of supplies needed to support instructional activities. There is a material list at the beginning of some units. Examples of lists of supplies found at the beginning of a unit include: 

  • Algebra 1, Unit 1: The following materials are listed: Helpful to create graphs (if you have a Mac): Omni Graph Sketcher (free), Desmos, Three-Act Task. 

  • Algebra 2, Unit 5: The materials listed include: Equations, tables, graphs, and contextual situations. A calculator or other technology to graph and solve problems using exponential modeling and logarithms.

  • Geometry, Unit 5: The material listed is the Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet.

There are times when the materials list is not comprehensive and/or omitted. Examples include, but are not limited to:

  • Algebra 1, Unit 4, Lesson 4, Target Task asks students to graph coordinates, connect coordinates to create a linear function, and then find the inverse; however, there are no materials listed for Unit 4 and no mention of materials needed in the lesson.

  • Algebra 1, Unit 4, Lesson 10, Anchor Problem 3 requires students to “write and graph a system of inequalities” but there are no materials listed in the unit or lesson.

  • Algebra 2, Unit 4, Lesson 8, Criteria for Success has reference to [TABLE] and [TBLSET], both functions on a graphing calculator. In addition, Anchor Problem 2 has these questions, “How can you use your graphing calculator to see possible differences?”, “Show the table in your graphing calculator for each of these functions using the [TABLE] feature. Which values of x are of most interest to you? Why?”, and “Why do both functions return an “ERROR” for the value at x = 2? Is the reason for the error the same? Different?” Unit 4 has no material(s) list nor is a graphing calculator mentioned as a needed material in the lesson. 

  • Geometry, Unit 6, Lesson 3, Target Task Part b requires students to find how much it will cost to paint the gym floor if cans of paint are $15.97. The only material listed for Unit 6 is the Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet. There is no direction if calculators should be utilized or not.

Indicator 3g

This is not an assessed indicator in Mathematics.

Indicator 3h

This is not an assessed indicator in Mathematics.

Criterion 3i - 3l

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

3/10
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA do not meet expectations for Assessment. The materials partially include assessment information that indicate which standards and practices are assessed and partially provide assessments that include opportunities for students to demonstrate the full intent of course-level standards.The materials do not provide multiple opportunities throughout the courses to determine students’ learning or sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3i

Assessment information is included in the materials to indicate which standards are assessed.

1/2
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Indicator Rating Details

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.

Indicator 3j

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

0/4
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The Assessment system does not provide multiple opportunities to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up with students.

The assessments for the materials include a post assessment after every unit and Target Task(s) at the end of each lesson. These provide little guidance to teachers for interpreting student performance or suggestions for follow-up. In Algebra 1 and Geometry, there is an Answer Key for each Post-Unit Assessment with point values assigned for each question. However, there are no rubrics or other explanations as to how many points different kinds of responses are worth. An example of this includes, but is not limited to:

  • Algebra 1, Unit 8: For Post-Unit Assessment question 6, students are given the following question: “Jervell makes the correct claim that the function below does not cross the x-axis. Describe how Jervell could know this and show that his claim is true.” The answer key states the following: “The discriminant of the quadratic formula tells how many real roots a quadratic function has (or how many times a parabola intersects with the x-axis). Since the discriminant of this function is − 32, there are no real roots to this function. (Equivalent answers acceptable)” This is worth 3 points, but there is no guidance for and no sample responses for which 0 points, 1 point or 2 points might be assigned. 

In the Target Tasks, there are no answer keys, scoring criteria, guidance to teachers for interpreting student performance, or suggestions for follow-up.

Indicator 3k

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

2/4
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Indicator Rating Details

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?”

Indicator 3l

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

Narrative Evidence Only
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Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

According to Math Teacher Tools, Assessment Resource Collection, “ The post-unit assessment is designed to assess students’ full range of understanding of content covered throughout the whole unit...It is recommended that teachers administer the post-unit assessment soon, if not immediately, after completion of the unit. The assessment is likely to take a full class period.” While all students take the post-unit assessment, there are no accommodations offered that ensure all students can access the assessment.

Criterion 3m - 3v

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

5/8
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Criterion Rating Details

The materials reviewed for Fishtank Math AGA do not meet expectations for Student Supports. The materials provide: extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity and provide manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The materials partially provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning course-level mathematics. The materials do not provide strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics.

Indicator 3m

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

0/2
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning series mathematics. There are no strategies, supports, or resources for students in special populations to support their regular and active participation in grade-level mathematics.

The materials have Special Populations under the Math Teacher Tools link. Within Special Populations, there is a link to Areas of Cognitive Functioning. Eight areas of cognitive functioning: Conceptual Processing, Visual Spatial Processing, Language, Executive Functioning, Memory, Attention and/or Hyperactivity, Social and/or Emotional Learning and Fine Motor Skills, are discussed in this section. While these areas of cognitive functioning are discussed in relation to mathematics learning, there are no specific suggestions and/or strategies for how teachers can assist students with their learning, if presented with these behaviors. Found in the Overview for the section on Areas of Cognitive Functioning, there is a statement that says: “To learn more about how teachers can effectively incorporate strategies to support students in special populations in their planning, see our Teacher Tools, Protocols for Planning for Special Populations and Strategies for Supporting Special Populations.” However, the protocols and strategies teacher tools  are not available in Fishtank Math AGA.

Indicator 3n

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

2/2
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity.

Opportunities for students to investigate course-level mathematics at a higher level of complexity are found with the lesson Anchor Problems. Each lesson contains Anchor Problems that are accompanied by Guiding Questions. The Guiding Questions assists students with critically engaging in the math content of the problem. Also, Guiding Questions prompt students to engage in purposeful investigations and extensions related to the problem. Examples of lessons that include the use of Guiding Questions for prompting students to engage in lesson content at higher levels of complexity include:

  • Algebra 1, Unit 2, Lesson 19: In Anchor Problem 1, students use screenshots of a battery charge indicator to determine when a laptop will be fully charged. Students need to represent the data in a scatter plot, determine the correlation coefficient for this data to determine the strength of the association, assign a line of best fit either through least squares regression or estimation, and determine if a linear function is the best model for this data through plotting the residuals. Guiding Questions that accompany this problem include:

    • How do the correlation coefficient and the residual plot help you to assess the validity of the answer to the question?

    • Why is it useful to have a line of best fit for this problem? How does this allow you to make a prediction?

    • How can you communicate your confidence in your answer to the question using correlation and the residual plot?

  • Algebra 1, Unit 5, Lesson 16: For Anchor Tasks Problem 2, students use the Desmos activity, Transformations Practice, and are tasked to write an equation that represents the blue graph for each transformation. At the end of the activity, there are two challenge transformations for students to complete. Guiding Questions for this problem include:

    • How can you tell if a reflection is involved?

    • How can you tell if a dilation or scaling of the graph is involved?

    • How can you tell if a translation of the graph is involved?

    • How are these moves represented in the equation?

  • Algebra 2, Unit 4, Lesson 18: Anchor Problem 1 involves two participants in a 5-kilometer race. The participants’ distances are modeled by the following equations: a(t)=\frac{t}{4} and b(t)=\sqrt{2t-1} where t represents time in minutes. Students need to determine who gets to the finish line first? Guiding Questions for this problem include:

    • What is the time for each person when the total distance run is 5 kilometers?

    • How can you use this information to determine who wins the race? 

    • If the participants have a constant speed, how many different times would you expect they would be side by side? 

    • How would you determine at what time(s) the participants are side by side? 

Sometimes teachers are directed to create problem sets for students that engage students in mathematics at higher levels of complexity.  Examples include:

  • Geometry, Unit 7, Lesson 2: The lesson objective is: Given a circle with a center translated from the origin, write the equation of the circle and describe its features. For the Problem Set, teachers are to create the problems for students. Teacher directives for creating the problems include three bullet points labeled EXTENSION. These bullet points read as follows:

    • Include problems such as “What features are the same/different between the two circles given by the equations: x^2+y^2=16 and 2x^2+2y^2=16? Show your reasoning algebraically.”

    • Include problems with systems of equations between two circles, which is discussed in Algebra 2.

    • Include problems such as “What are the x-intercepts of the circle?”

  • Algebra 2, Unit 1, Lesson 12: The objective for this lesson is: Write and evaluate piecewise functions from graphs. Graph piecewise functions from algebraic representations. For the Problem Set, one of the directives to teachers states include problems: “where based on a description of number of pieces, continuous or discontinuous, students create a piecewise function graphically and algebraically (This is an extension, and we’ll come back to this at the end of the unit.)”  

  • Algebra 2, Unit 2, Lesson 2: The lesson objective is: Identify the y-intercept and vertex of a quadratic function written in standard form through inspection and finding the axis of symmetry. Graph quadratic equations on the graphing calculator. For the Problem Set, teachers create the problems and one of the directives to teachers is: “Include problems where students are challenged to write multiple quadratic equations given the constraint of vertex AND y-intercept in standard form. Ask students to explain what they have discovered about the possible values.”

Additionally, there are no instances of advanced students doing more assignments than their classmates.

Indicator 3o

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA partially provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials provide a variety of approaches for students to learn the content over time. Each lesson has Anchor Problems/Tasks to guide students with a series of questions for students to ponder and discuss, and the Problem Set, gives students the option to select problems and activities to create their own problem set. The Tips for Teachers, when included in the lesson, guides teachers to additional resources that the students can use to deepen their understanding of the lesson. However, while students are often asked to explain their reasoning, there are no paths or prompts provided for students to monitor their learning or self-reflect.

Indicator 3p

Materials provide opportunities for teachers to use a variety of grouping strategies.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA partially provide opportunities for teachers to use a variety of grouping strategies.

Some general guidance regarding grouping strategies is found within the Math Teacher Tools, Academic Discourse section, however there is limited guidance on how to group students throughout the Fishtank Math AGA materials. Grouping strategies are suggested within lessons, however these suggestions are not consistently present or specific to the needs of particular students. Occasionally, there will be some guidance in the Tips for Teachers on how to facilitate a lesson, but this is limited and inconsistent. Examples include:

  • In Algebra 1, Unit 8, Lesson 13: In the Tips for Teachers, there is a bullet point that states, “There is only one Anchor Problem for this lesson, as there is a lot to dig into with this one problem. Students can also spend an extended amount of time on independent, pair, or small-group practice working through applications from Lessons 11–13.” While grouping students is suggested, no guidance is given to teachers on how to group students based on their needs.

  • In Geometry, Unit 6, Lesson 6: Problem 3 Notes state: “Students should spend time discussing and defending their estimates before being given the dimensions of the glasses. Students should first identify information that is necessary to determine a solution and ask the teacher for this information, which can be given through the image “The Dimensions of the Glasses.” However, no particular grouping arrangement is mentioned.

Indicator 3q

Materials provide 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.

1/2
+
-
Indicator Rating Details

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.

Indicator 3r

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide a balance of images or information about people, representing various demographic and physical characteristics. No images are used in these materials. However lessons do include a variety of problem contexts that could interest students of various demographic populations. Examples include:

  • Algebra 1, Unit 1, Lesson 4: Any student could relate to Anchor Problem 1: “You are selling cookies for a fundraiser. You have a total of 25 boxes to sell, and you make a profit of $2 on each box.” In Lesson 5, Anchor Problem 3: “You leave from your house at 12:00 p.m. and arrive to your grandmother’s house at 2:30 p.m. Your grandmother lives 100 miles away from your house. What was your average speed over the entire trip from your house to your grandmother’s house?” 

  • Algebra 1, Unit 7, Lesson 13: There is a link in the Problem Set to Engage NY Mathematics: Algebra 1,Module 4,Topic A,Lesson 9, Exercise 3, Example 3: “A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion of the ball could be described by the function: h(t)= -16t2^+144t+160 where 𝑡 represents the time the ball is in the air in seconds and h(t)represents the height, in feet, of the ball above the ground at time 𝑡. What is the maximum height of the ball? At what time will the ball hit the ground?” Students graph the function and use the graph to determine problem solutions.

Names used in problem contexts are not representative of various demographic and physical characteristics. The names used can typically be associated with one population and therefore lack representation of various demographics. Examples include, but are not limited to:

  • Algebra 1, Unit 3, Lesson 7: Anchor Problem 1 begins with: “Joshua works for the post office and drives a mail truck.”

  • Algebra 2, Unit 1, Lesson 1: Anchor Problem 3 begins with: “Allison states that the slope of the following equation is 3.” In Lesson 3: Anchor Problem 2 begins with: “Alex is working on a budget after getting a new job.”

  • Geometry, Unit 8, Lesson 2: Anchor Problem 2 begins with: “Dan has shuffled a deck of cards.”

Other names found in the materials that are not representative of all populations include: Mary, Beverly, Andrea, Lisa, Greg, and Jessie.

Indicator 3s

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide guidance to encourage teachers to draw upon student home language to facilitate learning. There is no evidence of promoting home language knowledge as an asset to engage students in the content material or purposefully utilizing student home language in context with the materials.

Indicator 3t

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning. Within the About Us, Approach, Culturally Relevant, the materials state, “We are committed to developing curriculum that resonates with a diversity of students’ lived experiences. Our curriculum is reflective of diverse cultures, races and ethnicities and is designed to spark students’ interest and stimulate deep thinking. We are thoughtful and deliberate in selecting high-quality texts and materials that reflect the diversity of our country.” Although this provides a general overview of the cultural relevance within program components, materials do not embed guidance for teachers to amplify students’ cultural and/or social backgrounds to facilitate learning.

Indicator 3u

Materials provide supports for different reading levels to ensure accessibility for students.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide supports for different reading levels to ensure accessibility for students. There are no supports to accommodate different reading levels to ensure accessibility for students. The Guiding Questions, found within the lessons, offer some opportunities to identify entrance points for students. However, these questions provide teacher guidance that may or may not support struggling readers to access and engage in course-level mathematics.

Indicator 3v

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

2/2
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

While there are missed opportunities to use manipulatives, there is strong usage of virtual manipulatives such as Desmos and Geogebra throughout the materials to help students develop a concept or explain their thinking.They are used to develop conceptual understanding and connect concrete representations to a written method. Examples of the usage of virtual manipulatives include:

  • Algebra 1, Unit 1, Lesson 10, Anchor Problem 2, in the notes Teachers are instructed to show students a video of three people eating popcorn at different rates. The notes states that, “This video is essential to show students so they can graph this scenario accurately. You will likely need to show it several times.” 

  • Algebra 2, Unit 9, Lesson 1: The Problem Set contains a link to the Desmos activity, “Domain and Range Introduction.”

  • In Geometry, Unit 3, Lesson 10, Anchor Problem 2, animation in Geogebra is used for students to describe the transformation(s) that map one figure onto the other figure. 

Opportunities for students to use manipulatives are sometimes missed as the materials provide pictures but do not prescribe manipulatives. An example of this includes, but is not limited to:

  • In Algebra 2, Unit 8, Lesson 1, Anchor Problem 2, a picture of a spinner is shown, no physical or virtual spinner is provided. Cubes are mentioned in Anchor Problem 3, but there are no suggestions as to how to make simple cubes. In the Target Task, Game Tools listed include a spinner and a card bag, but there are no suggestions to teachers to provide these manipulatives.

Criterion 3w - 3z

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

0/0
+
-
Criterion Rating Details

The materials reviewed for Fishtank Math AGA integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in series standards. The materials do not include or reference digital technology that provides opportunities for collaboration among teachers and/or students. The materials have a visual design that supports students in engaging thoughtfully with the subject, and the materials do not provide teacher guidance for the use of embedded technology to support and enhance student learning.

Indicator 3w

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the series standards, when applicable.

DESMOS is used throughout the materials and it is customizable as teachers can copy and change activities or completely design their own. Examples include:

  • Algebra 1, Unit 2, Lesson 17: Tips for Teachers encourages teachers to use Desmos to help students understand Regressions. 

  • Algebra 1, Unit 5, Lesson 12: Tips for Teachers explains: “Desmos activities are featured in these lessons in order to capture the movements inherent in these transformations”.

  • Algebra 2, Unit 1, Lesson 12: The Problem Set contains a link to a DESMOS activity where students explore Domain and Range of different functions. 

Examples of other technology tools include:

  • Algebra 1, Unit 2, Lesson 6: Contains a link in the Problem Set to an applet with which students can explore Standard Deviation.

  • Algebra 2, Unit 8, Lesson 11: Contains a link in the Teacher Tips to a “Sample Size Calculator” that can be used to determine the sample size needed to reflect a particular population with the intended precision.  

  • Geometry, Unit 1, Lesson 2: Tips for Teachers contains links to Math Open Reference “Constructions”, and an online game called “Euclid: The Game” designed with Geogebra that assists students in understanding geometric constructions. 

  • Geometry, Unit 6, Lesson 10: In Tips for Teachers the following suggestion is made: ”The following GeoGebra applet may be helpful to demonstrate Cavalieri’s principle, which can be done after Anchor Problem #1: GeoGebra, “Cavalieri’s Principle,” by Anthony C.M OR.”

Indicator 3x

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

While there are opportunities within activities in this series for students to collaborate with each other, the materials do not specifically include or reference student-to-student or student-to-teacher collaboration with digital technology.

Indicator 3y

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within the units and lessons that supports learning on the digital platform. Each lesson contains the following components: Lesson Objective, Common Core Standards, Foundational Standards, Criteria for Success, Anchor Problems, and Target Tasks. In addition to these components, most lessons contain Tips for Teachers and Problem Set links.

While there is a consistent design within the units and lessons that supports learning on the digital platform, this design mainly supports teachers by giving guidance for lesson presentations and providing links to learning resources. There are no separate materials for students. Student versions of the materials have to be created by teachers.  While the visual layout is appealing, there are various errors within the materials. Examples include, but are not limited to:

  • Algebra 1, Unit 1, Lesson 8, Anchor Problem 1 has a link to a video of a ball rolling down a ramp so that students can sketch a graph of the distance the ball travels over time; however, the YouTube video says it is unavailable and is a private video. Also, in Anchor Problem 2, the fourth bullet under Guiding Questions is incomplete: “The equation that represents a quadratic function is. How can you verify the points you created on the graph using this equation?”

  • Algebra 1, Unit 5, Lesson 13, Anchor Problem 2, the first and second questions under Guiding Questions have an equation and then the word “{{ h}}ave” following it. The brackets should not be in either question.

  • Algebra 2, Unit 7, Lesson 13, Problems Set, two of the three links do not work. The first one gives an “Error 404 - Not Found” when clicked and the third link says “Classzone has been retired.”

  • Geometry, Unit 5, Lesson 8, Anchor Problem 3, under Notes, there is a link to NCTM’s Illuminations. However, when clicked, a “Members-Only Access” page appears.

Indicator 3z

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

Narrative Evidence Only
+
-
Indicator Rating Details

The materials reviewed for Fishtank Math AGA do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable. While teacher implementation guidance is included for Anchor Problems/Tasks, Notes, Problem Set, and Target Task, there is no embedded technology within the materials.

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Report Published Date: 2022/01/12

Report Edition: 2021

Please note: Reports published beginning in 2021 will be using version 1.5 of our review tools. Version 1 of our review tools can be found here. Learn more about this change.

Math High School Review Tool

The high school review criteria identifies the indicators for high-quality instructional materials. The review criteria supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our review criteria evaluates materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The High School Evidence Guides complement the review criteria by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways. 

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom. 

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.

Math K-8

  • Focus and Coherence - 14 possible points

    • 12-14 points: Meets Expectations

    • 8-11 points: Partially Meets Expectations

    • Below 8 points: Does Not Meet Expectations

  • Rigor and Mathematical Practices - 18 possible points

    • 16-18 points: Meets Expectations

    • 11-15 points: Partially Meets Expectations

    • Below 11 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 38 possible points

    • 31-38 points: Meets Expectations

    • 23-30 points: Partially Meets Expectations

    • Below 23: Does Not Meet Expectations

Math High School

  • Focus and Coherence - 18 possible points

    • 14-18 points: Meets Expectations

    • 10-13 points: Partially Meets Expectations

    • Below 10 points: Does Not Meet Expectations

  • Rigor and Mathematical Practices - 16 possible points

    • 14-16 points: Meets Expectations

    • 10-13 points: Partially Meets Expectations

    • Below 10 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 36 possible points

    • 30-36 points: Meets Expectations

    • 22-29 points: Partially Meets Expectations

    • Below 22: Does Not Meet Expectations

ELA K-2

  • Text Complexity and Quality - 58 possible points

    • 52-58 points: Meets Expectations

    • 28-51 points: Partially Meets Expectations

    • Below 28 points: Does Not Meet Expectations

  • Building Knowledge with Texts, Vocabulary, and Tasks - 32 possible points

    • 28-32 points: Meet Expectations

    • 16-27 points: Partially Meets Expectations

    • Below 16 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 34 possible points

    • 30-34 points: Meets Expectations

    • 24-29 points: Partially Meets Expectations

    • Below 24 points: Does Not Meet Expectations

ELA 3-5

  • Text Complexity and Quality - 42 possible points

    • 37-42 points: Meets Expectations

    • 21-36 points: Partially Meets Expectations

    • Below 21 points: Does Not Meet Expectations

  • Building Knowledge with Texts, Vocabulary, and Tasks - 32 possible points

    • 28-32 points: Meet Expectations

    • 16-27 points: Partially Meets Expectations

    • Below 16 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 34 possible points

    • 30-34 points: Meets Expectations

    • 24-29 points: Partially Meets Expectations

    • Below 24 points: Does Not Meet Expectations

ELA 6-8

  • Text Complexity and Quality - 36 possible points

    • 32-36 points: Meets Expectations

    • 18-31 points: Partially Meets Expectations

    • Below 18 points: Does Not Meet Expectations

  • Building Knowledge with Texts, Vocabulary, and Tasks - 32 possible points

    • 28-32 points: Meet Expectations

    • 16-27 points: Partially Meets Expectations

    • Below 16 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 34 possible points

    • 30-34 points: Meets Expectations

    • 24-29 points: Partially Meets Expectations

    • Below 24 points: Does Not Meet Expectations


ELA High School

  • Text Complexity and Quality - 32 possible points

    • 28-32 points: Meets Expectations

    • 16-27 points: Partially Meets Expectations

    • Below 16 points: Does Not Meet Expectations

  • Building Knowledge with Texts, Vocabulary, and Tasks - 32 possible points

    • 28-32 points: Meet Expectations

    • 16-27 points: Partially Meets Expectations

    • Below 16 points: Does Not Meet Expectations

  • Instructional Supports and Usability - 34 possible points

    • 30-34 points: Meets Expectations

    • 24-29 points: Partially Meets Expectations

    • Below 24 points: Does Not Meet Expectations

Science Middle School

  • Designed for NGSS - 26 possible points

    • 22-26 points: Meets Expectations

    • 13-21 points: Partially Meets Expectations

    • Below 13 points: Does Not Meet Expectations


  • Coherence and Scope - 56 possible points

    • 48-56 points: Meets Expectations

    • 30-47 points: Partially Meets Expectations

    • Below 30 points: Does Not Meet Expectations


  • Instructional Supports and Usability - 54 possible points

    • 46-54 points: Meets Expectations

    • 29-45 points: Partially Meets Expectations

    • Below 29 points: Does Not Meet Expectations