High School - Gateway 1

The materials reviewed for Mathspace High School Traditional Series meet expectations for attending to the full intent of the mathematical content 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.

Examples of standards addressed by the courses of the series include:

• N-RN.3: In Algebra 1, Subtopic 1.01, Lesson, Example 1, students show the product of 3 and 0.\bar{1} is rational. In Example 3, students show the sum of two rational numbers is rational by rewriting x and y as \frac{a}{b} and \frac{c}{d}, respectively, finding a common denominator, and adding the fractions. In Example 4, students prove the product of rational number -3 and irrational number \sqrt{2} must be irrational by using contradiction. In Example 5, students explain why the sum of an irrational number and a nonzero rational number is irrational.

• N-CN.2: In Algebra 2, Subtopic 2.03, Lesson, Example 4, students simplify expressions involving complex numbers and justify each step using the commutative, associative, and distributive properties. 

• A-REI.2: In Algebra 2, Subtopic 5.05, Lesson, students determine whether given values are viable solutions, extraneous solutions, or not solutions to given equations, identify how viable and extraneous solutions are identified on graphs of rational functions, and solve rational equations identifying which solutions, if any, are extraneous. In Subtopic 7.03, Lesson, students solve radical equations and identify whether solutions are viable or extraneous. 

• F-IF.4: In Algebra 1, Subtopic 3.04, Practice, students identify, interpret, and describe key features of functions represented in graphs and tables, as well as create graphs to represent contextual situations. In Subtopic 3.08, Lesson, Example 1, students graph and describe the key functions of a piecewise function.  In Subtopic 5.01, Lesson, students describe key characteristics of exponential equations and use the key components of exponential functions to create graphs. In Subtopic 10.01, Lesson, students identify and interpret key features of quadratic functions given a real-world context, graph a quadratic function showing its key features and identify when a quadratic function has zero, one, or two real solutions using its graph. In Algebra 2, Subtopic 6.04, Practice, Worksheet, Questions 14 and 15, students identify the period of a sine function from graphs related to sound and tempo.

• G-SRT.4: In Geometry, Subtopic 7.04, Lesson, students explore the relationship between the sides of a triangle when a line is drawn parallel to a third side using dynamic software followed by additional examples applying the conjectures developed during the exploration. In Subtopic 8.01, Lesson, Example 2, students derive the Pythagorean Theorem given a right triangle with an altitude drawn to its hypotenuse.

• G-GPE.6: In Geometry, Subtopic 10.02, Lesson, students explore how to divide a horizontal line segment according to given ratios. In examples 4 and 5, students divide segments according to a given ratio using similar triangles. Teachers guide students to generalizing this method for dividing any segment into a given ratio.

• S-ID.5: In Algebra 1, Subtopic 8.01, Engage, students design and conduct a survey, representing the data in a two-way frequency table, and describe the findings. In the Lesson section of the lesson, students represent given data sets with two-way frequency tables and interpret relative frequencies.

The materials reviewed for Mathspace High School Traditional Series meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. Materials intentionally develop the full intent of the modeling process throughout the series leading to culminating experiences that address all, or nearly all, of the modeling standards.

Throughout the series there are a total of 12 Subtopics identified as modeling, four in each course. Eleven of the 12 Subtopics focus on non-plus modeling standards. Many of the modeling Subtopics include an Engage which introduces students to the concepts being addressed. The Lesson component then provides the steps of the modeling process and examples for students to understand what modeling entails. Students are provided Practice via worksheets which include modeling problems. Throughout these modeling Subtopics, students experience the full intent of the modeling process with nearly all of the modeling standards. 

The following examples use the full intent of the modeling process: 

• In Algebra 1, Subtopic 5.04, Engage, students investigate R_{0} values, and graph and identify key features of exponential functions. Students choose one of the listed countries and model the number of infected people for the first twelve days of the disease assuming that on day zero only one person was infected. Students create a function based on the number of days passed, identify the kind of function produced by their model, graph the model, and provide at least one additional representation. Students also identify any assumptions and limitations of their chosen model. Students share their work with the class during a gallery walk and then prepare with their group a discussion that describes at least two new situations and/or factors that would change the shape of each graph. (F-IF.4, F-IF.5, F-IF.8b, F-IF.9, F-LE.1c and F-LE.5)

• In Algebra 1, Subtopic 8.04, Engage, students view a nutrition fact label and list questions that they could ask about it. Students then explore the calories versus macronutrients scatter plots in the applet and familiarize themselves with the context terms and function. Students create two new nutrition facts labels that meet certain nutritional and caloric targets, and use the scatter plots to summarize the relationship between calories and each of the macronutrients. Pairs of students present to the whole class before facilitating a class discussion. In the discussion guide, it is suggested after the presentation to provide the whole class an opportunity to revisit their work and make a revised estimation for each of the macronutrient values after seeing presentations on the ways different groups produced their estimates. Students can then reflect on the following prompt: “If you were to estimate the macronutrient values for a 100 calorie menu item, what method would you use to determine the macronutrient values? Is this the same method you used in the Explore? Why or why not?” (S-ID.6, S-ID.7 and S-ID.8) 

• In Geometry, Subtopic 9.06, Practice, Worksheet, Question 2, students use the provided image of a child’s toy to design a 3D-printed image of it given certain constraints.  Students create a model for the solid of revolution that would fit the removed portion of the model, determine possible dimensions for the diameter of the removable center that will create the toy, but keep the amount of plastic within certain constraints. Students make a recommendation about how the toy should be designed, including a model with the necessary dimensions and an estimate of how much plastic will be needed to print the toy. Students also provide an explanation as to why their dimensions are the best choice. (G-MG.1, G-MG.2 and G-MG.3)

• In Algebra 2, Subtopic 4.04, Practice, Worksheet, Question 1, students design a model for a roller coaster that falls within the average heights and lengths of roller coasters and contains at least 3 peaks. Students define variables and parameters and state any restrictions. Students explain how they created their model and justify why the model they created is a good model for a roller coaster. (A-SSE.1, A-SSE.1a, A-SSE.1b, A-REI.11 and F.BF.1)

• In Algebra 2, Subtopic 8.04, Practice, Worksheet, Question 1, students construct at least two models using different functions to represent data and explain which function is best. Students analyze and use their models to justify why the life expectancy increasing at a decreasing rate makes sense and explain the implications of life expectancy growing at different rates. (A-CED.2, F-IF.4, F-IF.5, F-IF.6, F-IF.9 and F-BF.1)

The materials reviewed for Mathspace High School Traditional Series meet expectations for, when used as designed, allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

Examples of how the materials allow students to spend the majority of their time on the WAPs include : 

• In Algebra 1, Subtopic 3.09, students create linear functions to model a real-world situation or data set. When creating a graph, students decide the independent and dependent variables and must choose a scale that shows the domain and range. Students choose an appropriate level of accuracy when reporting quantities. In Subtopic 4.05, students create systems of inequalities to represent constraints from a real-world context. Students modify their modeled solution to reflect new domain and range restrictions in order to make a solution viable in context. (N-Q.1, N-Q.2 and N-Q.3) 

• In Algebra 1, Subtopic 9.05, students factor trinomials using grouping to create equivalent expressions. In Algebra 2, Subtopic 3.03, students prove and use polynomial identities to rewrite polynomial expressions. In Subtopic 3.04, students compare the process of long division of real numbers to long division of polynomials by a monomial to generalize a process for polynomial long division and recognize the structures involved. In Subtopic 3.05, students apply various strategies to factor polynomials to reveal information about the roots. In Subtopic 3.06, students use structure and algebraic methods to find zeros and factors for polynomial equations. (A-SSE.1, A-SSE.2 and A-SSE.3) 

• In Algebra 1, Subtopic 3.04, Engage, students choose a graph and a context, writing a story that connects the key features of the graph to their chosen context.  In Subtopic 10.02, Engage, students graph a quadratic function, interpret key features in context, and create a business plan based on the information they obtain from the graph. In Algebra 2, Subtopic 1.01, Engage, students construct a graph of a rollercoaster by combining pieces of functions and analyze the key features. (F-IF.4, F-IF.5, F-IF.6 and F-IF.7)

• In Algebra 1, Subtopic 7.04, Engage, students compare the scores of two softball teams by analyzing and interpreting the shape, center, and spread of the provided box plots to determine which team will be more difficult to defeat. In Subtopic 7.04, Practice, Worksheet, Question 7, students determine which plant produces more beans on average and which plant has a more consistent yield based on the number of beans picked for 10 days from two different plants. (S-ID.2 and S-IC.1) 

• In Geometry, Subtopic 7.01, students dilate a figure given a scale factor, identify the scale factor used to dilate a figure ,and explore the properties of dilations. In Subtopic 7.02, students determine whether figures are similar by performing similarity transformations to map one figure onto another  including sequences of rigid transformations. In Subtopic 7.03, students look at the special cases of triangles and develop properties that help determine triangle similarity. Students are able to prove two triangles are similar and write proofs involving similar triangles. (G-SRT.A and G-SRT.B)

• In Geometry, Subtopic 8.01, students explore relationships of right triangle similarity to prove the Pythagorean Theorem by deconstructing a right triangle into two similar right triangles and exploring side length proportions. In Subtopic 8.03, the trigonometric ratios for right triangles are defined. Students use diagrams and interactive software to explore the relationships between these ratios. By the end of Geometry, Topic 8 Triangle Trigonometry, students make sense of, model, and solve mathematical and contextual problems involving right triangles by determining which trigonometric equations to apply based on the structure of the triangles and given values. (G-SRT.C)

The materials reviewed for Mathspace High School Traditional Series, when used as designed, meet expectations for allowing students to fully learn each non-plus standard. However, the instructional materials for the series, when used as designed, do not enable students to fully learn a few of the non-plus standards.

Examples of the non-plus standards that would not be fully learned by students when using the materials as intended include:

• A-SSE.3c: In Algebra 1, Subtopic 5.02, Lesson, Example 1b, students rewrite f(x)=2^{x-1} as f(x)=\frac{1}{2}(2)^{x} in order to identify the y-intercept and growth rate. In Subtopic 5.03, Lesson, Example 1b, students rewrite f(x)=4^{-x} as f(x)=(\frac{1}{4})^x in order to identify the y-intercept and rate of decay. Students have limited opportunity to use the properties of exponents to transform expressions for exponential functions.

• F-IF.8a: In Algebra 1, Subtopic 10.02, Lesson, Teacher guide, Exploration, students determine if two functions are equivalent, and then identify and interpret the intercepts of a given function in the context of a water balloon thrown by a child from a low diving board. In Practice, Worksheet, Question 18b, students graph a quadratic function in factored form that models a person jumping off a springboard into a diving pool and make a prediction of where the diver will enter the water. In Subtopic 10.03, Lesson, Example 3, students write a quadratic function for a rock thrown into a lake given a y-intercept and a maximum height after a certain period of time. In Example 4, students rewrite a quadratic equation in vertex form by completing the square, identify the vertex, the vertex as a maximum or minimum, and sketch the graph of the parabola. However, opportunities to interpret zeros, extreme values, and symmetry of quadratic functions in context are limited.

• S-ID.6a: In Algebra 1, Subtopic 8.04, Lesson, students calculate the line of best fit and correlation coefficient for a given set of data, determine when an exponential model is a better fit, use software to find regression equations, and interpret the meaning of the function in context. In Subtopic 8.05, Practice, Worksheet, Question 4, students look at a graph of a residual plot to determine whether a linear model is an appropriate choice for the data. For Questions 14b and 15b students construct a residual plot for the data and state or describe whether a linear model will be suitable for the data set. In Subtopic 10.06, Practice, Worksheet, Question 4, students determine the type of function that would accurately model the information presented in a table. However, the materials provide students limited opportunities to specifically fit quadratic models to data.

The materials reviewed for Mathspace High School Traditional Series meet expectations for requiring students to engage in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from Grades 6-8.

Examples of age-appropriate contexts throughout the series include:

• In Algebra 1, Subtopic 2.02, Engage, students are given equations containing the size and prices of two chosen solar panel systems. When using the Geogebra applet, students “balance the variables” with their chosen size and cost of system 1 and system 2. Students then determine the amount of months of use needed for both systems to cost the same. 

• In Geometry, Subtopic 3.04, Example 7, students find the coordinates for the location of a fountain so that it will be equidistant from the paths along the perimeter of a triangular garden. In Subtopic 3.06, Practice, Worksheet, Question 9, students determine the perimeter of a  shaded region and write an expression for the area of a belt buckle design involving multiple triangles, some shaded and some not. 

• In Algebra 2, Subtopic 4.03, Practice, Worksheet, Question 19, students are given a polynomial equation modeling the path of a dolphin leaping in and out of the water. Students graph the polynomial modeling the path of the dolphin and then explain whether or not they think the polynomial is a good model for the dolphin's path.

The materials regularly use various types of real numbers in Geometry and Algebra 2; however, Algebra 1 has a limited amount of number types used. Examples of the materials using various types of real numbers include:

• In Algebra 1, Topic 4, students solve systems of equations with coefficients and solutions that include decimal values and whole numbers.

• In Geometry, Subtopic 8.01, students use the Pythagorean Theorem and its converse to solve problems with rational and irrational solutions. 

• In Algebra 2, Subtopic 3.06, students solve polynomial equations with solutions that are rational, irrational, and complex.

Examples of applying key takeaways from grades 6-8 include:

• In Algebra 1, Subtopic 7.01, Practice, Worksheet, Question 20, students apply understandings from 6.SP.4 to explain the advantages of using a dot plot to display the data given representing judges’ scores. Students also explain the advantages or disadvantages of displaying the data on a stem-and-leaf and make a conjecture about the winning contestant based on the data. (S-ID.1) 

• In Geometry, Subtopic 7.01, students apply knowledge of scale factors (7.G.1) and dilations (8.G.3) to compare the effects of the dilation when the center of dilation is on a line segment of the pre-image to when the center of dilation is not on the pre-image. (G-SRT.1) In Subtopic 9.07, Lesson, students apply these concepts to dilations of three-dimensionals shapes and the effects on each dimension. (G-GMD.3)

• In Algebra 2, Subtopic 9.01, students apply understandings of 7.SP.1 and 7.SP.2 to engage with strategies including experimental, observational, and survey methods of statistical design. Students also describe when each method is appropriate in answering a statistical question. (S-IC.3)

The materials reviewed for Mathspace High School Traditional Series meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The following examples are instances where meaningful connections are made within courses:

• In Algebra 1, Topic 2, students solve linear equations and inequalities. In Subtopic 3.02, students learn characteristics of domain and range of discrete, continuous, and step functions. (F-IF.1 and F-IF.5) In Subtopic 3.04, students interpret key features of functions including domain, range, x-intercept, average rate of change, end behavior, positive interval, negative interval, and interval of decrease. (F-IF.4) In Subtopic 5.01, students learn key features specific to exponential functions. (F-IF.4, F-IF.7e and F-IF.9) In Topic 6, students connect the patterns of change between arithmetic sequences and linear functions, and geometric sequences and exponential functions, and distinguish between the two types of patterns. (F-IF.3, F-IF.4, F-IF.5, F-IF.6 and F-IF.9) In Subtopic 10.01, students learn key features specific to quadratic functions. (F-IF.4, F-IF.5 and F-IF.6) In Subtopic 10.05, students compare key features of linear, exponential, and quadratic functions. (F-IF.9)

• In Geometry, Subtopic 1.01, students expand their geometric knowledge of objects and use precise notation to describe geometric figures. (G-CO.1) In Subtopic 1.03, students define and construct copies of line segments and segment bisectors. In Subtopic 1.04, students define and construct copies of angles and angle bisectors. (G-CO.12) In Subtopic 2.02, students use angle relationships to prove lines are parallel and construct lines that are parallel. (G-CO.9 and G-CO.12) In Subtopic 3.06, students define and construct the midsegment of a triangle and apply theorems about midsegments to solve problems involving triangles. (G-CO.10 and G-CO.12) In Subtopic 11.04, students construct a square inscribed in a circle and apply properties of inscribed angles to prove theorems. (G-CO.13 and G-C.3)

• In Algebra 2, Subtopic 5.01, students learn that members of a family of functions share the same type of rate of change, and that this characteristic of rate of change determines the kinds of real-world phenomena that the function in the family can model. (F-BF.1a, F-BF.1b and F-BF.3) In Subtopic 5.06, students explore how the sum of a finite geometric series is the sum of all of the terms from a finite geometric sequence and how it is useful when making predictions and solving problems involving real-world situations that show an exponential relationship. (A-SSE.4) In Subtopic 5.07, students use rational functions to model a variety of real-world situations, including those that involve inverse variation. (F-IF.8)

The following examples are instances where meaningful connections are made throughout the series: 

• In Algebra 1, Subtopic 3.05, students graph linear functions using slope and interpret rate of change in context. (F-IF.6 and F-IF.7) In Subtopic 6.06, students distinguish between linear and exponential functions based on their rate of change. (F-IF.6, F-IF.9 and F-LE.1) In Subtopic 8.04, students write the equation of a linear model that fits bivariate data and interpret the slope in context of the data. (S-ID.6a and S-ID.7)  In Geometry, Subtopic 10.03, students prove theorems about parallel and perpendicular lines in the coordinate plane using slope. (G-GPE.5) In Algebra 2, Subtopic 1.02, students identify functions families for given equations and contexts based on the type of rate of change. (F-IF.6)

• In Algebra 1, Subtopic 4.01, students solve systems of linear equations by graphing. In Subtopic 4.02, students solve systems of linear equations by substitution. In Subtopic 4.03, students solve systems of linear equations by elimination. (A-REI.6) In Subtopic 11.06, students solve linear-quadratic systems by graphing and substitution. Students also interpret the solution(s) in context. (A-REI.7) In Geometry, Subtopic 11.07, Practice, Worksheet, Question 17, students determine the epicenter of an earthquake by solving a system of three quadratic equations (A-REI.C). In Algebra 2, Subtopic 8.03, students solve systems of linear and nonlinear equations. (A-CED.3, and A-REI.11) 

• In Algebra 1, Subtopic 7.03, students extend their knowledge of variability to the standard deviation and investigate the effects of outliers on data displays. In Subtopic 7.04, Lesson, Example 1, students interpret the exercise habits of two people based on the shape, center, and spread of each person’s exercise data. In Example 2, students create a display for a data set, interpret the data in context based on the shape, center, and spread. Then, students identify the outlier of the data set and describe the effects of its removal from the set on the shape, center, and spread. (S-ID.1, S-ID.2 and S-ID.3) In Algebra 2, Subtopic 9.03, students extend their knowledge of how the mean and standard deviation of a data set affect its distribution to develop the idea of a normal distribution. Students investigate variations of the bell curve when the standard deviation is small and large. Students use the empirical rule to determine probability of events and use z-scores to compare male and female height restrictions for an amusement park. (S-ID.4) In Subtopic 9.04, Lesson, Example 1, students determine that a sampling distribution of apple weights is approximately normal using the mean and standard deviation of the distribution. Students predict the margin of error for a phone’s battery life given the mean and standard deviation of a sample of 25 phones. (S-IC.4) In Subtopic 9.05, Lesson, Example 2, students determine if a new diet for a farmer’s chickens affects egg weights using a sampling distribution of 180 randomizations. Students determine if claims that the new diet increases egg weight are reasonable by determining the margin of error of a sample of hens on the new diet compared to a confirmed population mean weight on the old diet. (S-IC.6)

The materials reviewed for Mathspace High School Traditional Series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards.

Previous standards can be found in three locations within the teacher materials. In the “Materials Guide”, under the “Textbook Guides” section, the “Correlations and Alignment” document lists previous standards under “Prior Connections.” In the Topic Overview, previous standards are listed as “Foundational Knowledge.” Within the Subtopic Overview, they are referenced as “Prior Connections.” 

The introduction in the student materials does reference previous knowledge, however, specific standards are not listed.

Examples where the teacher materials explicitly identify content from Grades 6-8 and allow students to extend their previous knowledge include: 

• In Algebra 1, Subtopic 1.04, students extend their knowledge of the properties of integer exponents (8.EE.1), square roots of perfect squares and cube roots of perfect cubes (8.EE.2), and decimal expansions of rational and irrational numbers (8.NS.1) to rewrite radicals using exponents and exponents using radicals. (N-RN.1 and N-RN.2)

• In Algebra 1, Subtopic 2.01, students create and solve linear equations in both mathematical and real-world contexts (A-CED.1 and A-REI.3), while justifying the steps in the process to see if the solution is true. (6.EE.5)

• In Algebra 1, Subtopic 9.03, students expand expressions using prime factorization (7.EE.1) by making connections between the different forms of numbers (7.EE.2 and 8.EE.1). Students extend their knowledge by learning how to use the Greatest Common Factor (GCF) to factor polynomial expressions. (A-SSE.2 and A-SSE.3) 

• In Geometry, Subtopic 3.01, students extend their knowledge of angles sums within a triangle, the definition of exterior angles, and angles formed by parallel lines cut by a transversal (8.G.5) by proving the triangle angle sum theorem and the exterior angle theorem. (G-CO.10)

• In Geometry, Subtopic 8.02, students extend their knowledge of solving right triangles by using the Pythagorean Theorem (8.G.7) to find the ratio of side lengths in 45-45-90 and 30-60-90 triangles (G-SRT.6 and G-SRT.8).

• In Algebra 2, Subtopic 5.03, students extend their knowledge of multiplication and division of rational numbers (7.NS.2) to rewrite rational expressions using multiplication and division. (A-SSE.2) 

• In Algebra 2, Subtopic 5.05, students extend their knowledge of writing and solving equations that include non negative rational number(6.EE.7) and multiplying and dividing rational number(7.NS.2) by solving and graphing rational equations with viable and extraneous solutions.(A-CED.4, A-REI.2, F-IF.4 and F-IF.8)

• In Algebra 2, Subtopic 9.02, students extend their knowledge of developing probability models (7.SP.7) by executing a probability simulation to create theoretical and experimental probability models and compare the result to understand how fair decisions are made. (S-IC.2)