High School - Gateway 2

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).

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).

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)

The 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.

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).

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).

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).