High School - Gateway 1

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials address all aspects of all non-plus standards across the courses of the series.

Some examples of non-plus standards that were addressed by the series include:

• N-RN.2: In Algebra 2, Unit 3, Lessons 3 and 4, students engage with rational exponents. During Lesson 3, students rewrite rational exponents as radical expressions. In Lesson 4, students use rational exponents and the properties of exponents learned from integer exponents. In Activity 4.2 Students explain why $$(5^{\frac{1}{3}})^2$$ is equivalent to $$(5^2)^\frac{1}{3}$$. Students then rewrite both of the terms as radicals.

• N-CN.1: In Algebra 2, Unit 3, Lesson 10, students begin to develop an understanding of the imaginary unit. In Lesson 11, students evaluate expressions that result in imaginary numbers and plot those imaginary numbers on a coordinate plane. In Activity 11.4, students develop a complex number using a coordinate plane with a real number axis and an imaginary axis.

• A-REI.4a: In Algebra 1, Unit 7, Lesson 12, students solve quadratic equations by completing the square. In the warm-up, students make an argument of why $$x^2+10x+20$$ is not a perfect square. The teacher notes explain the reason $$\sqrt20$$ is not an option. In the lesson, students complete a table that starts with factoring and work backward from factored to polynomial form, allowing students to use the work at the top of the table to inform their work at the bottom. Students use two examples to solve equations by completing the square. Students compare and contrast the methods and solve several equations themselves. Later, in Lesson 19, students complete and examine the steps of deriving the quadratic formula. Students discuss the steps to understand how the quadratic formula relates to completing the square.

• F-IF.7a: In Algebra 1, Unit 6, Lesson 6, students graph quadratic functions that represent physical phenomenon, interpret key features of the graph in the real-world context given in the problem, and write and interpret quadratic functions that represent physical phenomenon. The teacher notes say: “Students use a linear model to describe the height of an object that is launched directly upward at a constant speed. Because of the influence of gravity, however, the object will not continue to travel at a constant rate (eventually it will stop going higher and will start falling), so the model will have to be adjusted (MP4). They notice that this phenomenon can be represented with a quadratic function, and that adding a squared term to the linear term seems to “bend” the graph and change its direction.” After guided exploration, the students answer the question: “Why do you think the graph that represents $$d=10+46t$$ changes from a straight line to a curve when $$-16t^2$$ is added to the equation?” Through the guided exploration using technology, students interpret different parts of the function, the vertex of the graph of the function, and the zeros of the function.

• G-CO.9: In Geometry, Unit 1, there are three lessons that address this standard. In Lesson 19, students critique a conjecture which states that an angle formed between angle bisectors is always a right angle. Students proceed to develop their own conjecture about vertical angles, which leads to the proof that vertical angles are congruent. In Lesson 20, students translate and rotate one of two intersecting lines to produce parallel lines cut by a transversal, and they prove theorems related to the angles formed by parallel lines cut by a transversal. In Lesson 21, the Triangle Sum Theorem is proven in two different ways using transformations. 

• G-GPE.1: In Geometry, Unit 6, Lesson 4, the three activities support students in deriving the equation of a circle given the center and radius using the Pythagorean Theorem. Students use the Pythagorean Theorem to calculate segment length, test whether a point is on a circle, and apply that perspective to build the general equation of a circle. The three activities in Geometry, Unit 6, Lesson 6, also support students in completing the square to find the center and radius of a circle. Students complete perfect square trinomials, analyze a demonstration of completing the square, and complete the square to find the center and radius of a circle. 

• S-ID.4: In Algebra 2, Unit 7, Lesson 6, students find the area under a curve and connect this concept to using the mean and standard deviation to describe the proportion of the data in an approximately normal distribution. In Algebra 2, Unit 7, Lesson 7, students use the area under a normal curve to find the proportion of values in certain intervals and have the option to extend their learning by finding an interval that fits a certain percentage of the data. 

• S-ID.9: In Algebra 1, Unit 3, Lesson 9, Activity 9.2, students distinguish between correlation and causation given different situations. Students look for relationships between the scatterplots and the words and have a discussion about causation and correlation. In Lesson 9, Activity 9.3, students describe situations that exhibit varying degrees of causal relationships. Students determine if situations have a very weak (or no) relationship, a strong relationship that is not causal, or a causal relationship.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials use the full intent of the modeling process to address nearly all of the modeling standards across the courses of the series.

Each course in the series provides modeling prompts in the teacher materials, which are separate from the units and lessons. The modeling prompts include multiple versions of a task, sample solutions, and instructions for the teacher around implementing the modeling task in the classroom. There is also guidance provided about the best time to use each prompt (e.g., “Use after Unit 4, Lesson 8”). The second task statement for each prompt typically provides some scaffolding and removes some obstacles that might prevent all students from accessing the material in the first task statement. Students engage in parts of the modeling process with the alternate prompts throughout the series.

Examples where the full intent of the modeling process is used to address modeling standards across the courses of the series include:

• In Algebra 1, Modeling Prompt, “Giving Bonuses” (N-Q), students determine how bonuses should be distributed among the five workers who worked on a project. Students create a proposal with at least two different methods to distribute the bonuses (formulate) to present to their boss along with a recommendation for a specific method with an argument in support of the recommended method. Students compute each employee’s bonus (compute) and provide advantages and disadvantages for each method proposed (interpret). Students also discuss which of the five employees would be most likely to complain about each method and how they would justify the method to the employee (validate).

• In Algebra 1, Modeling Prompt, “Planning a Concert” (N-Q, A-CED.2, A-REI.4), students propose a cost for concert tickets based on provided survey data (formulate/compute). Students consider the cost of the band, the cost of a venue, and revenue that might be generated through the sale of concessions and merchandise and how these aspects impact the cost of a ticket (interpret/compute). The students create and present a plan that includes information about the cost and the profit based on information about the venue, expenses, and ticket prices (validate/report).

• In Geometry, Modeling Prompt, “A New Container” (G-GMD.3, G-MG.1,3), students design a new container that must hold a volume of 16 fluid ounces. Students use any three-dimensional shape or a combination of shapes (formulate) to create a design that is appealing and meets the criteria for volume. They provide a model or prototype of their container and the calculations that prove that the container will hold 16 fluid ounces (compute/interpret/validate). The students present their designs and defend their mathematics to others (reporting). Each student/group is able to have different shapes/designs, but all presentations are considered valid if the criteria have been met.

• In Geometry, Modeling Prompt, “So Many Flags” (G-SRT.8, G-MG.1,3), students create a flag for Nepal, of any size, using provided instructions about its construction. Students decide the sizes of large and small flags for a parade of flags (formulate). Students compute the measure of each of the angles in the flag they have constructed, the amount of material needed to create the flags, and the amount of ribbon needed to sew ribbon along the border of each flag.

• In Algebra 2, Modeling Prompt, “How Big Is That?” (N-Q, A-CED.A, G-GMD.3, G-MG), students write part of a children’s book that compares the sizes of different animals by relating their size to other objects that might be familiar (formulate). Students determine how they will scale the objects to communicate the different sizes of the objects being compared and how they will scale the smaller object in order to make an appropriate comparison (compute/interpret/validate). Students present a portion of their book as well as the relevant mathematics to justify their work (validate/report). Each student/group has a choice in what they are going to compare. All results are considered valid if there is evidence that students completed the task based on the criteria.

• In Algebra 2, Modeling Prompt, “Swing Time” (N-Q.2,3, S-ID.6) students formulate conjectures about variables that might affect the period of a pendulum and validate their conjectures by collecting data and determining if the data makes sense in the problem. Students formulate models to demonstrate the relationship between the identified variable and period of the pendulum. Students complete computations to predict different “timed” periods that cannot be performed in class. Students revisit their work and make any necessary changes after they decide how many significant digits to use. Students report their findings throughout the task by explaining and justifying their decisions.

There are also examples of modeling scenarios within the lessons. For example, in Algebra 1, Unit 2, Lesson 1 (A-CED.2), students estimate how much a pizza party would cost. Students write expressions to show how they arrived at their estimate. Students consider many factors and identify quantities in their expressions that could change on the day of the party. Students formulate expressions to represent the parameters of the pizza party, make assumptions during this process, and set constraints. Students also compute the cost of the party based on their estimates and parameters. Groups consider how the estimated costs would change if parameters changed, allowing students to interpret their results and make adjustments to their initial estimates.

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs and careers. Examples of how the materials allow students to spend the majority of their time on the WAPs include:

• N-RN.A: In Algebra 2, Unit 3, Lesson 3, students apply knowledge of exponential equations and rules of exponents to develop understandings of how rational exponents are related to equivalent radical expressions. In Lesson 4, students rewrite fractional exponents as a unit fraction times a whole number and rewrite the expressions using radicals, and connect roots, rational exponents, graphs of exponential functions, and decimal approximations. In Lesson 5, students further develop their understanding to include rational exponents, rules of exponents, and graphs to make sense of negative rational exponents (N-RN.1,2). In Algebra 2, Unit 4, Lessons 4, 6, and 7, students interpret fractional inputs for exponential functions in context. They also use properties of exponents to interpret and transform expressions that represent decay, and students use fractional exponents to answer questions about amounts of radioactive isotopes in old artifacts. (N-RN.1)

• A-SSE: In Algebra 1, Units 5, 6, and 7 address many of the standards in A-SSE. For example, in Algebra 1, Unit 5, Lesson 9, students recognize and discuss similarities and differences in $$x^2$$ and $$2^x$$ as the beginning of interpreting exponential functions. Students interpret different parts of the exponential function in a real-world scenario (A-SSE.1,2). In Algebra 1, Unit 6, Lesson 8, students “explain why the diagram shows that $$6(3 + 4) = 6\cdot3 + 6\cdot4$$”. Students “draw a diagram to show that 5(x + 2) = 5x + 10.” The lesson continues using structure to find equivalent quadratic expressions (A-SSE.2,3). In Algebra 2, Unit 2, Lesson 3, students are introduced to polynomial functions and use graphing technology to write polynomials given specific characteristics (A-SSE.1,2).

• F-IF: In Algebra 1, Unit 4, students interpret and use function notation, analyze and create graphs of functions, find the domain and range of functions, and find, write, and interpret inverse functions. Students extend their work with F-IF standards in Algebra 2, Unit 2, Lesson 1 as they construct an open box and calculate the volume. Students find the largest volume, write an expression, and use graphing technology to create the graph. Students use their knowledge of the key features of graphs and domain/range to create their box. 

• G-SRT: In Geometry, Unit 3, Lesson 13, Activity 13.3, students use similarity criteria to write statements indicating why the three triangles in the picture provided are similar. Additionally, in Lesson 15, Activity 15.2, students complete an activity involving task and data cards while engaging in discussion about what information is needed and why it is needed to solve the problems about triangle similarity (G-SRT.5). This is prerequisite work for G-SRT.8.

• S-ID.2: In Algebra 1, Unit 1, Lesson 5, students calculate interquartile range and discuss the importance of outliers. Students also find the Mean Absolute Deviation (MAD) and use their understanding of the MAD to interpret given scenarios. Students create two different sets of six data points which could be possible locations of pennies along a meter stick that would result in a given MAD. In Lesson 11, Cool Down, students use the interquartile range and/or the MAD to compare the spread of four data sets by examining menu prices to determine the best menu based on the greatest variability in menu options relative to price. In Lesson 12, students use the MAD to develop standard deviation by learning what happens to the standard deviation (derived through technology) when specific numbers are manipulated as in the following: the lowest value is removed; the greatest value is removed; the greatest value is doubled, etc. 

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for, when used as designed, letting students fully learn each non-plus standard. The instructional materials for the series, when used as designed, enable students to fully learn all of the non-plus standards. Examples of how the materials allow students to fully learn all of the non-plus standards include:

• N-CN.7: In Algebra 2, Unit 3, Lessons 17, 18, and 19, students solve quadratics using the quadratic formula and by completing the square when solutions are complex. This standard is addressed in all three lessons as well as in the practice assignments. Students continue to engage in solving equations with complex solutions in practice sets throughout Algebra 2, Unit 3.

• A-APR.6: In Algebra 2, Unit 2, Lesson 12, students divide polynomials by linear factors using area models traditionally used to support the multiplication of polynomials. The activities include division that results in a remainder. In Lesson 13, students use long division to divide polynomials, and in Lesson 15, students engage with The Remainder Theorem. In Lesson 19, students apply long division to rewrite rational expressions in order to reveal the end behavior of the function.

• A-REI.4a: In Algebra 1, Unit 7, students have multiple opportunities to complete the square and derive the quadratic formula. In Lesson 12, students recognize perfect square expressions and build perfect square trinomials, and students develop the rule for completing the square. In Lesson 14, students examine visual models representing the process of completing the square and use “u-substitution” to complete the square. In Lesson 15, students find irrational solutions by completing the square. In Lesson 19, completing the square is further developed as students derive the quadratic formula. In Algebra 2, Unit 3, Lesson 16, students compare different methods for solving quadratic equations, including completing the square, and determine when it might be best to use each method. Practice sets for each lesson listed provide additional problems for students to practice completing the square. 

• F-IF.7b: In Algebra 1, Unit 4, Lesson 12, students graph piecewise functions in applications, such as cost for shipping related to weight and renting a bike for minutes used. Absolute value functions are addressed in Algebra 1, Unit 4, Lessons 13 and 14. In Lesson 13, students create a scatter plot of the absolute guessing error calculated from guesses for the number of objects in a jar. In Lesson 14, students graph absolute guessing error again for temperatures and work with the distance function (absolute value function). Students plot graphs by hand and by the use of technology. In Geometry, Unit 5, Lesson 5, students scale the area of different objects (floor area, painting area), graph the area with the scale factor, which results in a square root function, and explain the behavior of the graph. In Geometry, Unit 5, Lesson 7, students apply the same technique when scaling volume to produce the cube root function. In Geometry, Unit 5, Lesson 18, students complete a volume problem that relates the scaling of a balloon to its volume and surface area, which revisits square root and cube root functions.

• F-BF.2: In Algebra 2, Unit 1, over multiple lessons, opportunities for students to work with arithmetic and geometric sequences are presented. In Lesson 5, students represent sequences graphically, numerically, and in a table, and determine if a sequence is arithmetic or geometric. In Lesson 6, students match sequences with the appropriate recursive pattern and represent a given sequence recursively. In Lesson 7, students write recursive patterns/sequences and determine the terms of the sequence. In Lesson 8, students transform recursive sequences into explicitly defined sequences. In Lesson 9, students define given sequences both recursively and non-recursively. In Lesson 10, students compare arithmetic and geometric sequences in the context of real-world applications. 

• G-C.2: In Geometry, Unit 6, Lesson 14, students consider a triangle that is inscribed in a semi-circle and examine the slope of the chords, relating the hypotenuse to the diameter of the circle. In Geometry, Unit 7, Lesson 1, students define chords, central angles, and inscribed angles, considering each one’s relationship to the circle as a whole. Students examine angle measures and arc measures related to central angles and inscribed angles. Students also examine chords, similar triangles that are formed by chords, and diameters. In Geometry, Unit 7, Lesson 2, students work with inscribed angles, the relationships to the intercepted arcs, and how the “rules” for finding those angle measures are related to central angles. In the practice problems, students explain or describe the difference between central angles and inscribed angles. In Geometry, Unit 7, Lesson 3, students complete problems related to radii and segments through the points of tangency and use principles of perpendicular lines to solve problems. In Geometry, Unit 7, Lesson 14, students synthesize content from previous lessons by solving problems in the context of real-world scenarios, such as pizza (circles) and flashlight beams (arcs).

• S-ID.9: In Algebra I, Unit 3, Lesson 9, students explore the definitions of correlation and causation. Students examine real-world scenarios and make determinations as to how the data is correlated, as well as if there is a causal relationship between the variables. In Algebra I, Unit 3, Lesson 10, students have additional opportunities to consider both correlation and causation and explain their reasoning. 

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series 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 where the materials illustrate age-appropriate contexts for high school students include:

• In Algebra 1, Unit 3, Lesson 5, students compare pounds of ice cream sold to the outside temperature and rider service prices compared to distance traveled, as they use technology to write equations of lines. (S-ID.6, S-ID.7)

• In Algebra 1, Unit 6, Lesson 7, students work with quadratic functions to describe the number of downloads of a movie and how it impacts the revenue generated. (F-BF.1, F-IF.7)

• In Geometry, Unit 3, Lesson 16, students explore similar triangles to make a bank shot in a pool game. (G-SRT.5)

• In Geometry, Unit 8, Lesson 11, students play “Rock, Paper, Scissors” and discuss how different events influence the outcome. (S-CP.6) 

• In Geometry, Modeling Prompt 4, students discuss and determine their water usage daily and weekly during different tasks performed in everyday life. Students research and “describe a container that would hold the amount of water you use in a week, a month, a year, and a lifetime”. (G-GMD.3, G-MG.1, N-Q.1-3)

• In Algebra 2, Unit 4, Lesson 18 includes applications of exploring acidity and the corresponding pH scale, measuring the intensity of earthquakes, and calculating the balance in a bank account.

Examples where students apply key takeaways from Grades 6-8 include:

• In Algebra 1, Unit 4, Lesson 1, Activity 1.1, students apply their knowledge of functions (8.F.1) as they engage with contextual relationships that do and do not represent functions (F-IF.1). In Lesson 1, Activity 1.2, students use their understanding related to 8.F.5 as they reason graphically about the relationship between time and the distance of a dog from a post (F-IF.4). 

• In Algebra 1, Unit 5, Lesson 1, students apply ratios and proportional relationships (7.RP.A) to compare examples of linear growth and exponential growth while working on Activity 1.2 about a genie in a bottle. This builds to an understanding of exponential growth (F-LE.5, F-IF.4). 

• In Grade 8, students used similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane, and they derived the equation y = mx + b for a line intercepting the vertical axis at b (8.EE.6). In Geometry, Unit 6, Lesson 9, students develop the point-slope form of a linear equation: y - k = m(x - h). In ensuing lessons, students write equations of lines, and intercepts are not always readily available. (A-SSE.A, G-GPE.6)

• In Geometry, Unit 2, Lesson 1, Activity 1.2, students describe the sequence of transformations on figures in an open space instead of on a coordinate plane with exact numbers. Students apply their knowledge of rotations, reflections, and translations from Grade 8 to move figures and visualize the movements in “Are you ready for more?”. In this activity, students draw additional line segments in the given figures to make two congruent polygons (8.G.1,1b). In Lesson 1, Activity 1.4, students are offered “another opportunity to practice reasoning based on corresponding parts of figures they know to be congruent”. Using a reflection of triangles that share a side, students prove that the side shared is an angle bisector.

Examples where the instructional materials use various types of real numbers include:

• In Algebra 1, Unit 5, Lesson 4, students study exponential decay, and the numbers used are $$\frac{27}{4}$$ and $$\frac{81}{8}$$. Throughout the lesson, students work with fractions, and in Lesson 5, students work with decimals as they explore how much medicine remains in a patient's bloodstream over a period of time. 

• In Algebra 1, Unit 7, students work with radicals where some are rational and others are irrational. In Lesson 13, Activity 13.2, students complete the square with fractions and decimals as coefficients $$(x^2 +1.6x +0.63=0)$$. In Lesson 20, students add and subtract with integers, radicals, and fractions to decide if the sums will be rational or irrational.

• In Algebra 1, Unit 2, Lesson 5, students graph equations using large numbers (which need to be addressed through scale) and decimals. Practice Problem 5 also incorporates decimals to the hundredths place, and students consider limitations to domains as they solve application problems.

• In Geometry, Unit 5, Lesson 18, students determine the amount of helium, in cubic feet, needed to fill balloons of different sizes. These numbers get large (~21,000), so students would need to adjust their graphing calculator windows. 

• Algebra 2, Unit 3 includes two assessments on complex numbers and rational exponents. The Check Your Readiness assessment includes integers, integer and rational bases, radical expressions, and cube roots. The End-of-Unit Assessment extends to complex numbers.

 The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series 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 courses include:

• In Algebra 1, Unit 7, Lessons 20-21, students solve quadratic equations using a variety of methods (A-REI.4b) and defend whether or not the solutions are rational or irrational. Students investigate sums and products of rational and irrational numbers to develop general rules about the type of number the sums and products will be (N-RN.3). After completing both of these lessons, students determine if the solutions to quadratic equations are rational or irrational.

• In Geometry, Unit 6, Lesson 4, Activities 4.1 and 4.2, students subtract coordinates as part of a method for calculating the distance between two points, and they use the Pythagorean Theorem to test whether points are on a circle with a given center and radius (G-GPE.4). In Lesson 4, Activity 4.3, students apply their work to build the general equation of a circle (G-GPE.1). In Lesson 7, students apply their understanding of distances to parabolas based on the location of a focus and directrix. In the Lesson Synthesis, students compare and contrast the work completed in Lesson 7 with the work they did in Lesson 4. In Lesson 8, students build the equation for a parabola given a focus and directrix applying their understandings from previous lessons (G-GPE.2).

• In Algebra 2, Unit 5, throughout many lessons, students work with transformations of functions, both graphically and algebraically (F-BF.3). In Lesson 1, students examine a scatter plot of data for a cooling function and use the graph of the data and two given functions to determine which function best fits the data (S-ID.6a). The same data set is also presented in Lesson 7, where students describe how a given function can be translated to better fit the given data and write the function with the identified translations. The unit ends with students modeling given data by applying transformations to write functions that best fit the data. 

Examples where the materials foster coherence across courses include:

• In Algebra 1, Unit 7, Lessons 12, 13, and 14, students complete the square to solve quadratic equations (A-REI.4a). In Algebra 1, Unit 7, Lessons 22 and 23, students produce equivalent forms of quadratic expressions by completing the square to reveal properties of quadratic functions (A-SSE.3). Students use the skill of completing the square again in Geometry, Unit 6, Lessons 5 and 6, to write given equations of circles in standard form and to identify the center and radius of the circle (G-GPE.1). Completing the square is used again in Algebra 2, Unit 3, Lesson 17 to solve quadratic equations that include complex solutions.

• In Geometry, Unit 5, Lesson 7, students graph cube root functions while working backwards from the volumes of original and scaled solids to calculate scale factors (F-IF.7b). Students use the graph to analyze rates of change in the scale factor for different volume inputs. Students also graph cube root functions in Algebra 2, Unit 3, Lesson 2, as they reconnect the ideas of a square root representing a side length of a square and a cube root representing an edge length of a cube. 

• In Algebra 1, Unit 5, students write exponential equations and use context to compare linear and exponential models. Throughout the unit, students graph exponential functions and identify key components (F-IF.7e). In Algebra 2, Unit 4, students build on their understanding of exponential functions from Algebra 1 where students only worked with exponential functions with domains of integers. In Algebra 2, the domain is expanded to include all real numbers as students use exponential equations to model growth and decay (F-IF.4). 

The instructional materials reviewed for McGraw-Hill Illustrative Mathematics AGA series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials explicitly identify the standards from Grades 6-8 in the teacher materials. The Design Principles of the teacher materials state that the initial lesson in a unit is designed to activate prior knowledge and provide an entry point to new concepts. The lessons are organized in such a manner that each activity has a foreword that indicates standards by category: Building On, Addressing, and Building Towards, where appropriate. This information appears routinely in the design of the teacher materials but not in the student and family materials.

Examples where the teacher materials explicitly identify content from Grades 6-8 and build on them include: 

• In Algebra 1, Unit 1, Lesson 1, the Lesson Narrative of the preparation indicates that the work of the lesson builds on 6.SP.1, although there is no specific reference to grade 6 learning in the Lesson Narrative. The standard is indicated in the Building On portion of the CCSS Standard alignments. Students begin by identifying which of four given questions does not belong; the sample responses provided for the teacher indicate that students may respond that the questions are or are not statistical (6.SP.1). Students then develop survey questions based on three given statistical questions and survey the class to collect data. 

• In Algebra 1, Unit 1, Lesson 2, a connection is made to 6.SP.4 and the display of numerical data in plots on a number line, dot plots, histograms, and box plots. The materials state that this serves as a brief review of these representations and the way they are created prior to engaging in the work of S-ID.1 and S-ID.2 in ensuing lessons.

• In Algebra 1, Unit 2, Lesson 17, students build on their previous understanding of 8.EE.8 by considering systems of equations that have no solutions or infinitely many solutions. Students identify, without graphing or using algebra, if a system of linear equations is equivalent or parallel.

• In Geometry, Unit 5, Lesson 2, students build on 7.G.3, where they describe two-dimensional figures that result from slicing three-dimensional figures. Students analyze cross-sections of three-dimensional figures (G-GMD.4), and build toward G-GMD.1, where students identify three-dimensional solids given parallel, cross-sectional slices. 

• In Geometry, Unit 3, Lesson 3, Measuring Dilations (G-SRT.1) builds on 8.G.3. During this lesson, students dilate a quadrilateral using different scale factors. The purpose of this activity is to understand that the different ratios of the dilations are equal. 

• In Geometry, Unit 5, Lesson 16, students build on their previous understanding of 7.G.6 and 8.G.9 by solving surface area and volume problems with a real-world context. In the problems in the student materials, students maximize and minimize these geometric attributes which is an extension of previously learned skills in grades 7 and 8.

• In Algebra 2, Unit 3, Lessons 1-3 and 5, students build on previous knowledge of 8.EE. Student work extends beyond the rules of exponents that were learned in previous grades in the following ways: solving simple equations to find the missing exponents in an equivalent relationship; considering numbers expressed as square roots and determining which integers it falls between; considering unit fractions as exponents and how the rules of exponents extend to all rational numbers.