High School - Gateway 1

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.

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.

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.

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).

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).