High School - Gateway 2

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, students are expected to use multiple representations to further develop conceptual understanding. 

Examples of the development of conceptual understanding include:

• N-RN.1: In Alg2.4, Lesson 6.1, Warm-Up, students explain how given expressions with various exponents are equivalent and provide an additional equivalent expression. This helps students develop conceptual understanding of the properties of exponents.
• A-REI.6: In Alg1.2, Lesson 17, students determine solutions to a system of equations through inspection and use those solutions to determine that there are infinitely many solutions to the given system. Students recognize equivalent equations and explain what equivalence means in terms of solutions to systems of linear equations. Students also interpret what the solution of a system of equations would be if the equations represent parallel lines. 
• F-IF.2: In Alg1.4, Lesson 4, students match words to the symbolic rule of given functions. Students use symbolic notation to define the perimeter of a rectangle with a set height and varying width. Students graph the results and find both input and output values from the graph. In Alg1.4, Lesson 5, students expand this understanding to write functions that describe data plans for their phones and compare competing plans both symbolically and graphically. Students describe in writing how the graphic visualization matches the symbolic rule.
• G-GMD.1: In Geo.5, Lesson 13, students partition prisms in order to build the volume formula for a pyramid as opposed to using a given formula to calculate volume. Students connect the volume of a prism to the volume of a pyramid with a base area equivalent to that of the prism. 
• G-SRT.6: In Geo.4, Lesson 4, students connect angle measurements with ratios of side lengths in right triangles. In Lesson 6, students extend their thinking about the ratios of side lengths for any values of the triangle’s angles. Students define cosine, sine, and tangent and compare the answers they get using these definitions to the table used in the previous lessons.  
• S-ID.7: In Alg1.3, Lesson 4, students develop their understanding of slope while studying line of best fit related to a scatterplot. Students answer a series of questions to help develop their understanding of what happens to the slope of the line of best fit if one of the elements changed. An example is as follows: “How would the scatter plot and linear model change if grapefruits were used instead of oranges?” Additionally, students discuss the role of the y-intercept during this lesson by explaining what the y-intercept means in the particular context of the problem. Students repeatedly demonstrate understanding of the concept of slope and y-intercept in given data sets throughout the remainder of Algebra 1.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills and provide opportunities to independently demonstrate procedural skills throughout the series. The curriculum guide states, “We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.” Throughout the series, procedural skills are developed through the lessons and the problem sets for each of the lessons. Each problem set has cumulative practice problems to review previously addressed procedural skills. 

Examples that show opportunities for students to independently demonstrate procedural skills across the series include: 

• A-SSE.1: In Alg1.6 and Alg2.2, students develop procedural skill and fluency, as they make observations related to the structure of a factored quadratic expression and the zeros of that expression when graphed as a function. Students predict possible factors and their forms by evaluating the expression for varying input values, looking at graphs, reading tables, and exploring end behavior.
• A-APR.6: In Alg2.2, Lesson 17, students perform polynomial division by using long division, synthetic division, and polynomial factorization in order to write a higher-order polynomial as a product of its linear and/or non-linear factors.
• F-IF.1,2: In Alg1.4, Lessons 2 - 5, students use function notation and develop fluency with substitution and calculations. 
• F-IF.4: In Alg1.5 and Alg2.6, students develop procedural skill and fluency as they explore key features of multiple types of graphs. In Alg1.5, over the course of nine lessons, students build an understanding of how exponential growth differs from linear growth. They encounter different contexts and use expressions, graphs, and tables to distinguish between the two types of functions. They gain fluency in how to compare two exponential functions, how they differ in their expressions, and what that will mean for growth in context. In Alg2.6, students use these skills with trigonometric functions, recognizing and discussing amplitude, frequency, and shifts in many and varied contexts over the course of 4 lessons.
• F-BF.3: In Alg1.4, Lesson 14, students analyze the type of transformation a constant value creates with an absolute value function. In the cumulative practice problem set, there are more problems for students to practice this skill.
• G-GPE.4: In Geo.6, Lesson 10 practice, students write equations using parallel slopes and identify equations that are parallel to a specific line. In Geo.6, Lesson 11, students have the same opportunity with perpendicular slopes.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series 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 students utilizing mathematical concepts and skills in engaging applications include:

• A-REI.11: In Alg1.4, Lesson 9, students engage with two or more graphs simultaneously, interpreting their relative features and their average rates of change in context.  Examples are as follows: population, trends of phone ownership, and the popularity of different television shows. In Activity 9.2, students compare two functions by studying graphs and statements in function notation. In Alg2.4, Lesson 15.3, students solve a system of exponential equations involving a cicada population using logarithms and graphing; specifically, students explain why they can use the intersection of the two graphs to estimate when the cicada population will reach 100,000.
• F-IF.6 and F-BF.1: In Alg1.4, Lesson 18, students use functions to model real-life applications. Students create and analyze functions to model cell phone battery power using given data. In the cumulative practice, students engage with relevant applications related to distance driven over time and the relationship between temperature and cricket chirps.
• G-SRT.8: In Geo.4, Lesson 9.3, students use the safe ladder ratio to determine the safe ladder angle, and students use the calculated angle to decide if a ladder’s length is sufficient for a given scenario. Students also determine if it is possible to adjust the ladder to a safe angle and explain their reasoning. In Lesson 10, students solve application problems using trigonometry. Students find the perimeter of figures inscribed in a circle using trigonometric ratios, and  students solve problems involving an airplane’s angle of descent and path length as it descends to its destination.
• G-MG.2: In Geo.5, Lesson 17.3, students apply volume and density to determine the number of fish that could be housed in a tube-shaped aquarium with an open 4-foot cylinder in the middle for viewing.
• S-ID.6 and N-Q.3: In Alg1.3, Lesson 6, students apply residual value, line of best fit, and percent error to data related to the weight of oranges in a crate. Students also apply these concepts in practice problems 4 and 5 involving applications of car sales and temperatures.
• S-IC.1: In Alg2.7, Lesson 3, students evaluate the randomness in population samples. Students determine the best way to have random samples and the factors that could affect the randomness in several different scenarios. Students draw conclusions from a variety of non-routine application problems.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the materials, and multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. 

The following are examples of balancing the three aspects of rigor in the instructional materials: 

• Each lesson begins with a Warm-Up, and this is often an opportunity for students to develop their number sense or procedural fluency. After the Warm-Up, there are activities that do one or more of the following: provide context, introduce, formalize or practice vocabulary, work toward mastery, introduce a new concept, or provide an opportunity to model. The embedded classroom routines also contribute to a balance of the three aspects of rigor. These routines include the following:  Analyze It, Math Talks, Group Presentations, Notice and Wonder, Think and Share and others. At the end of the lesson, there is a synthesis activity where the teacher leads a discussion to formalize the learning. The lesson ends with a Cool-Down for students to work independently on the lesson concepts. Each of the aspects of rigor are addressed with this lesson structure throughout the series. 
• In Alg1.6, Lesson 9, students demonstrate a balance of the three aspects of rigor while addressing A-SSE.2,3. In the Practice Problems, students determine if two representations of quadratic expressions are equivalent. Students perform calculations related to quadratic functions in the context of real-world applications, one context of which is a football player throwing a pass. In the Student Task Statements, multiple representations of factorable quadratics are presented. Students examine standard form, factored form, and a geometric representation of a factorable quadratic expression. Students explain their reasoning when determining if an expression is in factored form. A-SSE.2 is also addressed in Geo.6, Lesson 5 when students apply the distributive property and squared forms of a binomial to derive the equations of circles. Students practice procedural skills using the distributive property and writing different forms of equivalent expressions. They also solve application problems related to equations of circles and distances. In Alg2.2 Lesson 23, A-SSE.2 is addressed by having students build conceptual understanding of polynomial identities. Through an application problem, students define an identity. Then, they multiply expressions to generalize patterns in polynomials, which develops procedural skill.
• In Alg1.1, Lessons 2 - 5, the materials provide activities that engage students in all aspects of rigor with respect to S-ID.2. Specifically, in Lesson 2.2, students represent and analyze histograms. In Lessons 3.1 and 4.1, students consider dot plots to inform a conversation about the shape of a distribution. In Lesson 5, students calculate the measures of central tendency. In Lesson 9, students perform statistical calculations. In ensuing lessons, students further develop statistical reasoning in the following lessons:  in Lesson 10, students consider what variables they may use to analyze a situation and describe data displays they may use to compare two sets of data; in Lesson 11, students explore and reason about symmetry in a data set; in Lesson 12, students investigate standard deviation and other measures of variability; and in Lesson 14, students investigate the effect of outliers. In Lessons 15 and 16, students compare measures of center and variability in context, as well as determine the best measure of center and variability for several data sets. Students also design an experiment to answer a statistical question, collect data, analyze data using statistics, and communicate the answer to the statistical question.
• In Geo.7, Lesson 6, students solve problems related to distance and parabolas. In the Student Task Statements, students answer questions related to the distance from the focus to the vertex of a parabola and the effect that distance might have on the shape of the parabola. They use the definition of a parabola and the distance formula to determine if a point is on the parabola. Students defend their answers and try to generalize how one would know if a point is on the parabola given a graph. Students also demonstrate an understanding of what happens to the shape of a parabola if one was to move the directrix closer to the focus.
• In Alg2.5, Lessons 8 and 9, students consider the impact of scaling the input or the output values of a function. They examine how graphs change based on the scaling of the input (horizontal) or output (vertical) values. In Lesson 9, Practice Problems, students determine if different statements that are made based on scaled inputs or outputs are correct given different representations of functions. Students also use data to determine an appropriate scale factor that would model the population of sloths given an initial function.
• In Alg2.2, Lesson 20.3, students write a simple rational equation about batting average to develop procedural skill, and they demonstrate conceptual understanding when working with a word problem and writing it algebraically. The extension and what-if questions about the rational equation address application of rational equations.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards.

Examples where students make sense of problems and persevere in solving them include:

• In many units throughout the series,  students answer the questions “What do you notice?” and “What do you wonder?”in some lesson activities. The goal of these questions is to guide classroom conversation toward the mathematical material that the class is about to address. These questions increase accessibility for students by providing entry points to the context, which aids in making sense of the tasks. 
• In Alg1.4, Lesson 1, students analyze the relationship between the number of bagels purchased and the cost of the bagels to determine how three different costs could all be true. 
• In Geo.4, Lesson 11, students examine how inscribed polygons with increasing numbers of sides can lead to an approximation of $$\pi$$. Students make sense of the problem to determine appropriate methods for finding a formula to calculate the perimeter of the inscribed polygons, which leads to an approximation of $$\pi$$.
• In Alg2.2, Lesson 24, students create multiple right triangles from a given set of instructions. Students also attempt to find one example that does not create a right triangle. Towards the end of the lesson, students develop an identity that can be used to generate Pythagorean Triples.

Examples where students attend to precision include:

• In Alg1.4, Lesson 6, students analyze a graph containing two mappings related to two objects in time. One graph shows a linear piecewise function produced by a drone, and the other graph shows a quadratic function produced by a toy rocket. Students describe the graphs’ representations in terms of the real-world contexts.
• In Geo.6, Lesson 7, as students articulate what they notice and wonder, they attend to precision in language they use to describe what they see. Students may initially propose less formal or imprecise language, then restate their observation with more precise language in order to communicate clearly. Relevant vocabulary includes the following: equidistant, congruent segments, and parabola.
• In Alg2.3, Lesson 6, students solve simple equations involving squares and square roots. The teacher notes state: “Students attend to precision when they reason about solutions to equations involving squares and square roots from the meaning of the √ symbol (MP6).” In this lesson, students explore the idea that every positive number has two square roots. The convention of giving only the positive root is also discussed in terms of its precise meaning. Students explore the use of the radical symbol as a tool of precision.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. Across the series, there is an intentional development of MP2 and MP3 that reaches the full intent of the MPs. There are many examples in the instructional materials of MPs 2 and 3 where students reason abstractly and quantitatively or construct viable arguments and critique the reasoning of others.

Examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Alg1.1, Lesson 14, students determine whether or not to exclude an outlier. In the Student Task Statements, Problem 14, students determine if there are outliers for a data set, explain why any outliers might exist, and determine if the outliers should be included in the analysis of the data.
• In Alg1.5, Lesson 10, students examine data from a cooling coffee function. They must determine, from given intervals, the best interval to use for an appropriate “average rate of cooling” and support their choice.
• In Alg1.7, Lesson 1.3, students write a quadratic equation and are prompted not to solve it. In writing an equation and interpreting the solution in its context, students practice reasoning quantitatively and abstractly.
• In Geo.3, Lesson 2, students create a scale model of the Solar System to verify the distance the Moon would be from the Earth when fully eclipsing the Sun.
• In Alg2.7, Lesson 16, students take two readings of their pulse. For one reading, they count the beats out loud while watching the clock, and for the second reading, they take a few deep breaths, close their eyes and have someone else watch the clock. They then compare the two rates. Data from the whole class is collected and a discussion held as they reason quantitatively and abstractly together.

Examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Alg1.1, Lesson 11.3, students explain their reasoning and critique the reasoning of others as they determine if a data display matches a written statement or not. In this activity, students have small group discussions and examine scenarios from a classmate’s point of view. Students also construct arguments they can defend for their own matches, as well as arguments for why they might disagree with their partner.
• In Alg1.7, Lesson 23.3, students explain why a vertex is a maximum or a minimum. Students explain their reasoning concerning which performance gives the greater maximum revenue without creating a graph. Students construct an argument, and during the class discussion, they critique the reasoning of others in the class.
• In Geo.2, Lesson 3, students construct arguments to describe that congruence through transformations requires a series of transformations where corresponding parts match each other. In the Cumulative Practice Problem Set, Problem 2, students argue for the congruence of two triangles based on a rotation and explain their reasoning, citing the argument for congruence based on the transformation.
• In Geo.6, Lesson 9, students construct viable arguments during the lesson synthesis by answering a question about which form of an equation of a line they prefer. In the previous lesson, students use equations in multiple forms to find out what the slope of the line is and what point each line passes through. Students explain why they prefer a specific form over another.
• In Geo.8, Lesson 2, students, in groups, draw slips of paper with a name on them from a bag with an unknown number of slips. They record the name, replace the slip, pass the bag, and draw again. After 15 draws, each student in the group makes predictions of how many names and how many slips were in the bag. If the group has consensus, they draw another round. Each time, they construct new arguments and critique the thinking of others.
• In Alg2.2, Lesson 4.3, students answer questions about operations on polynomials. They experiment to develop reasons to support their answers. During the group discussion, they defend and critique the reasoning of their classmates as they describe events as subsets through taking turns in trading roles as they explain their thinking.
• In Alg2.4, Lesson 14, students practice constructing logical arguments when they justify solutions and explain why a certain value is a reasonable estimate for a given logarithm.

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. There are multiple problems and activities throughout each unit in which students use and create mathematical models to enrich the mathematics. Students also choose from a variety of tools throughout the lessons, including digital tools provided in a drop-down menu in the online materials (MP5). 

Examples where students model with mathematics include:

• In Alg1.2, Lesson 20.2, students solve a problem about gas in a lawn mower. The teacher materials state, “To reason about the problem, students need to interpret the descriptions carefully and consider their assumptions about the situation. To make sense of the situation, some students may define additional variables or use diagrams, tables, or other representations. Along the way, they engage in aspects of modeling (MP4)”. Students identify the important quantities in this scenario, identify the relationships, and write at least one inequality to represent their conclusions.
• In Alg1.4, Lesson 17, students write a linear function for data concerning the percent of cell phones in homes in the United States since 2004. Students answer questions leading to finding a model for the inverse of the function. 
• In Geo.1, Lesson 9, students use perpendicular bisectors to decide which stores in the city will be responsible for orders based on the store’s location compared to where the order will be delivered. Students use real-world situations to apply their knowledge and make approximations on their calculations to simplify distributions for a local store.
• In Geo.7, Lesson 14, students work with a pizza slice as a model of a sector of a circle. Students compute the cost per square inch of pizza slices from four vendors by computing the sector area. This engages students in the modeling process by reporting their findings and considering other variables.
• In Alg2.2, Lesson 16, students engage in aspects of the modeling process by making reasonable estimations and determining reasonable constraints in the context of real-world scenarios. In “Are You Ready for More?”, students consider different aspects of manufacturing, other than simply minimizing materials, in order to make sense of an open-ended problem. 
• In Alg2.7, Lesson 14, students speculate whether the differences of the means in small experimental groups can be reduced by randomly regrouping the data. Students approximate the distribution of simulated differences of means by using a normal distribution. 

Examples where students choose and use appropriate tools strategically include:

• In Alg1.7, Lesson 17, students use the form of a quadratic equation as a “tool” to solve problems. Students “write quadratic equations to represent relationships and use the quadratic formula to solve problems that they did not previously have the tools to solve (other than by graphing). In some cases, the quadratic formula is the only practical way to find the solutions. In others, students can decide to use other methods that might be more straightforward.”
• In Alg1.5, Lesson 19.2, students compare linear and exponential growth involving simple and compound interest. Students strategically use technology, whether they make a graph (for which they will need to think carefully about the domain and range) or continue to tabulate explicit values of the two functions (likely with the aid of a calculator for the exponential function). 
• In Geometry, Unit 7, Lesson 7.2 students create an arbitrary triangle, use angle bisectors and constructions to find the incenter, and construct the triangle’s inscribed circle. The narrative states: “Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).” The narrative also states: “Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.”
• In Geo.8, Lesson 9 students use two-way tables as a sample space to decide if events are independent and to estimate conditional probabilities. Although technology is not required, it is recommended that technology be made available as there are opportunities for students to choose to use appropriate technology to solve problems. 
• In Alg2.4, Lesson 5, students create an exponential function given a table of values. During this lesson, teachers make sure that students have access to a spreadsheet tool to reason about the given questions. This helps students focus on the questions rather than the calculations, and students use tools to their advantage during the lesson. 
• In Alg2.5, Lesson 11, students apply transformations on functions to determine the best model for a data set, specifically temperature data from heating objects. The Lesson Narrative states: “This can be done by hand via experimenting, but students may also choose to use graphing technology to help choose the appropriate translations, scalings, and reflections (MP5).”

The instructional materials reviewed for LearnZillion Illustrative Mathematics Traditional series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. Each of these MPs is cited numerous times across the series, and the Algebra 1 Extra Support Materials cite each of these MPs. Additionally, across the series, the majority of the time MP7 and MP8 are used to enrich the mathematical content, and there is intentional development of MP7 and MP8 that reaches the full intent of the MPs.

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

• In Alg1.4, Lesson 12, students use piecewise notation to look for structure when graphing those functions. 
• In Geo.2, Lesson 13, students look for and make use of structure by working backward from the statement they are trying to prove about parallelograms.
• In Geo.3, Lesson 5, students use Notice and Wonder to examine triangles whose midpoints connect to form other smaller triangles. Students notice these smaller triangles are dilations of the larger triangle.
• In Geo.8, Lesson 9, students notice and make use of structure through a Math Talk as they recognize fraction bars as part of a fraction, and as representing division. 
• In Alg2.2, Lesson 8, students make a conjecture based on creating their own function and analyzing the end behavior to see if it matches their conjecture. The focus of the lesson is “using the structure of the expressions to understand how the term with the highest exponent dictates end behavior even when other terms may have larger values at inputs nearer to zero due to coefficients.” 
• In Alg2.7, Lesson 7, students find the area under the normal curve and interpret the proportion of values at different intervals. These problems utilize a real-world scenario, and students connect these applications to the theoretical study of the normal distribution. 

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

• In Alg1.1, Lesson 6, students experiment by changing numbers in a provided spreadsheet to discover what and how the number impacts the outputs. Students express regularity in repeated reasoning as they observe the outcome of different inputs to generalize the operations in the formula cell.
• In Alg1.2, Lesson 8, students repeatedly rewrite equations to isolate different variables. Students rely on previous knowledge of solving equations to generalize their reasoning as they work with literal equations. 
• In Alg1.6, Lesson 5.3, students look for repetition in their calculations for a falling object and how this relates to quadratic functions. The Teaching notes state, “To find a new expression that describes the height of the object, students reason repeatedly about the height of the object at different times and look for regularity in their reasoning (MP8)” as they write new quadratic expressions. Students also use repetition in their calculations as they complete provided tables to determine where an object would be at a certain time and use that information to write new expressions.
• In Geo.1, Lesson 6, students construct parallel and perpendicular lines, and students engage in MP8 as they repeatedly construct these different types of lines. 
• In Geo.7, Lesson 8, students perform multiple arc length and sector area calculations as they generalize formulas for each.
• In Alg2.1, Lesson 1, students look for a pattern within the Tower of Hanoi puzzle, where students complete a puzzle by building a tower. Students play the game and make some conjectures about the smallest number of moves you can make to complete the tower.
• In Alg2.1, Lesson 8, students use a table of values to generalize formulas for finding the nth term of a sequence. By examining patterns and applying repeated reasoning, students generalize the definition of a sequence into an equation and/or function.
• In Alg2.6, Lesson 10.3, students determine specific trigonometric values for large angles. “Students make connections between angles greater than 2$$\pi$$ and between 0 and 2$$\pi$$ that correspond to the same point on the unit circle (MP 8)”.