High School - Gateway 1

The instructional materials reviewed for the Mathematics Vision Project Integrated series meet the expectation that the materials attend to the full intent of the mathematical content contained in the high school standards for all students. Overall, the materials fully addressed the mathematical content of the standards, but there were a few instances where the materials failed to meet the full intent of the standard.

The following are examples of standards that are attended to fully by the materials:

• F-IF.3: In Secondary Math One, Module 2, Task 1 students work with arithmetic and geometric sequences including discrete and continuous linear and exponential situations. In Secondary Math One, Module 2, Task 2 students connect context with domain and distinguish between discrete and continuous functions, and in Secondary Math One, Module 3, Task 7 students identify whether or not a relation is a function given various representations.
• F-IF.7: This standard is addressed across multiple lessons and courses and incorporates a variety of functions in a variety of ways. For example, the standard is addressed in Secondary Math Two (2.2.1, 2.4.1-4) and Secondary Math Three (3.3.3,6,7,8).
• A-APR.1-5: These standards are addressed in Secondary Math Three, Module 3, Tasks 4-8. In task 4, students add, subtract, and multiply polynomials while looking for patterns and paying attention to end behavior. In task 5, students develop an understanding of multiplicity to gain a deeper understanding of the relationship between the degree and the number of roots of a polynomial. In task 6, students identify the degree of the polynomial, determine end behavior, use the Fundamental Theorem of Algebra, determine the multiplicity of a given root, and recognize graphs, including those with imaginary roots. In task 7, students apply the Remainder Theorem. In task 8, students factor, solve, and graph polynomials and find roots, determine multiplicity, and predict end behavior.
• N-RN and N-CN.1, N-CN.2, and N-CN.7: In Secondary Math Two, Module 3, Tasks 1-4, the introduction of rational exponents is done in context (i.e. bacteria population growth rates and interest on a savings account). Students choose their quantities and scale and explain why they are being used. When graphing, the students often begin with a blank grid and must supply the scale and labels they will use. In Secondary Math Two, Module 3, Task 9 students extend the real and complex number systems, and in Task 10 students examine the arithmetic of real and complex number systems, engaging students in the use of with rational and irrational numbers.
• S-CP.A: The materials address conditional probability in Module 9, Task 3 (using samples to estimate probabilities), Task 5 (examining independence of events using two-way tables), and Task 6 (using data in various representations to determine independence).

There are instances where the materials attend to part of the standard but do not attend to every aspect of the standard:

• G-GPE.7: Secondary Math One, Module 8, Tasks 1 and 3 address perimeter, but the materials do not address area. There are no tasks where students find area by using the coordinates.
• S-IC.4: The materials have students engage in the use of a random sample in the Ice Cream Task in Secondary Math Three, Module 9, Task 2, but population “mean” or “proportion” and “margin of error” are not mentioned. Also, the sample the task is analyzing is not truly “random,” because the participants in the survey are voluntary and not randomly selected.
• N-Q.3: Students are not required to choose a level of accuracy appropriate to limitations on measurement when reporting quantities. The standard is not listed in the “Core Alignment Document.” There are instances in the materials where students would need to choose a level of accuracy appropriate to the limitations on measurement when reporting quantities, such as Secondary Math Three, Module 6, when students are using Trigonometric functions to analyze the periodic rotation of the ferris wheel. Some of the answers will be irrational and require students to round and decide what place value would be best to round to. The materials do not appear to instruct students on how to make this decision.

These standards are not attended to by the materials:

• A-SSE.4: Students are not required to derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and then use the formula to solve problems such as calculating mortgage payments. The standard was listed in the “Core Alignment Document,” yet the reviewers did not find it addressed by any specific tasks.
• S-IC.5: The use of simulation as stated in the standard is not included in the series. The standard is listed in the “Core Alignment Document,” yet the reviewers did not find it addressed by any specific tasks.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. The materials provide opportunities for students to engage in every step of the modeling process. Tasks that involve modeling include a graphic of the modeling process in the teacher notes. Additionally, all the modeling standards are addressed in the materials.

Examples of modeling tasks include:

• Secondary Math One, Module 1, Task 2, “Growing Dots” addresses standards F-BF.1 and F-LE.1, 2, and 5. The students are presented with a visual pattern of dots and are asked to describe that pattern and to predict how the pattern would look after 3 minutes, 100 minutes, and "n" minutes. The teacher notes prompt the teacher to ask students to share out specific strategies and solution paths. While the teacher notes are scripted and prompt the teacher to seek out specific strategies, the problem leaves students open to any approach they find logical. The teacher notes place equal value on any of the possible student strategies and encourage students to analyze and discuss the variable strategies.
• Secondary Math One, Module 2, Task 5, “Making My Point” addresses standards A-SSE.1, A-SSE.2, A-CED.2, and F-LE.5. The focus is on understanding and using various notations for linear functions. Students are guided to create two different but equivalent equations that are a mathematical model of the context: quilt blocks in a quilting pattern. By constructing tables, drawing graphs, and completing patterns they investigate both equations in order to identify that they are in fact equivalent.
• Secondary Math Two, Module 1, Task 3, “Scott’s Macho March” addresses standards F-BF.1, F-LE.A, A-CED.1 and 2, and F-IF.4 and 5. Details about the number of push-ups Scott completes a day are provided, and students are left to interpret the information, formulate a strategy, and compute their answers. Students solve by extrapolating how the pattern will continue into the future as they are looking at the sum of the number of push-ups that Scott has completed on a particular day. The teacher notes provide instructions for teachers to have students share out their answers, interpret what their answers mean in context, and evaluate each other’s answers and strategies.
• Secondary Math Two, Module 7, Task 10, “Sand Castles” addresses standards G-GMD.1 and G-GMD.3. Students pretend they are entering a sand castle competition. Equipped with key parameters and details (like shape and size), this task has them analyze the area of the base and the volume of sand necessary for their three sand castles.
• Secondary Math Three, Module 5, “Modeling with Geometry,” and Task 4, “Hard as Nails” engage students in the modeling process. This task addresses standards G-MG.1, 2, and 3. In this task, the students are given a detailed drawing of a nail on a coordinate plane and are then asked to find the volume of an individual nail and then use that information to calculate how many nails would be necessary for a particular building project.
• Secondary Math Three, Module 6, Task 12, “Getting on the Right Wavelength” addresses standards F-TF.5 and F-BF.3 and 4. Students are given a picture of a Ferris wheel, along with a few details, and are then asked to write equations to model the height of the rider at any given time and make predictions about how the wheel will behave in the future.

While there are many examples of modeling problems throughout these materials, there are some problems labeled as “modeling” problems that provide scaffolding which inhibits students from engaging in the full modeling process.

The instructional materials reviewed for Mathematics Vision Project Integrated series, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, post-secondary programs, and careers. All of the WAP standards were addressed. Overall, the majority of the tasks addressed WAP standards. The percentage of tasks that addressed WAPs was the greatest in Secondary Math One and the least in Secondary Math Three. The Algebra and Function WAPs are emphasized to the greatest degree, followed by the Number and Quantity standards, while the Geometry and Statistics WAPs are given the least attention.

The WAPs from the Function Conceptual Category are included throughout the series. Evidence is found in Secondary Math One, Modules 1, 2, 3, 8; Secondary Math Two, Modules 1, 2, 4; and Secondary Math Three, Modules 1, 2, 3, 4, 6, 7.

The Algebra Conceptual Category standards are included throughout the series. Evidence is found in Secondary Math One, Modules 4, 5; Secondary Math Two, Module 1; and Secondary Math Three, Module 4.

The WAPs from the Geometry Conceptual Category are largely addressed in the Secondary Math One, Modules 6, 7, 8; Secondary Math Two, Modules 5, 6, 7, 8; and Secondary Math Three, Module 5.

The WAPs from the Statistics Conceptual Category are largely addressed in the Secondary Math One, Module 9; Secondary Math Two, Module 9; and Secondary Math Three, Module 8.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation for requiring students to engage in mathematics at a level of sophistication appropriate to high school. Students engage in investigations throughout each task that ground the standards in real-world context appropriate for high school use.

Some examples from various Modules and Tasks that highlight high school sophistication include, but are not limited to:

• Secondary Math One, Module 2, Task 6 “Form Follows Function”: This task builds fluency with linear and exponential functions by recognizing and efficiently using the information given in a problem. Students work with linear and exponential functions represented by a variety of tables, graphs, equations, and story contexts. This variety of representation, context and the numbers incorporated (annual rate of 2.4%, height in cm to the nearest tenth) make them appropriate for high school. (F-LE.2, F-LE.5, F-IF.7, A-SSE.6)
• Secondary Math One, Module 9, Task 7 “Getting Schooled”: Students are presented with the opportunity to analyze the Census Bureau’s income data to understand more about the differences in women’s and men’s salaries. Based on the data in this task and in Module 9, Task 6, “Making More $,” students make a case to support whether the difference in income may be explained by differences in education or discrimination and consider what other data would be useful. (S-ID.6, S-ID.7, S-ID.8)
• Secondary Math Two, Module 1, Task 5 “How Does It Grow?”: Students distinguish between relationships that are quadratic, linear, exponential, or neither. The materials include relationships presented with tables, graphs, equations, visuals, and story context. Students are asked to create a second representation for the relationships given. Graphing technology is recommended for this task. (F-LE.1, F-LE.2, F-LE.3)
• Secondary Math Two, Module 3, Task 5 “Throwing an Interception”: In this task, the quadratic formula is developed from the perspective of visualizing the distance the x-intercepts are away from the axis of symmetry, by engaging students in a scenario most high school students find relevant. Question 7 provides an alternative algebraic approach for deriving the quadratic formula that does not include completing the square. Instead, students make use of the idea that the x-intercepts are d units from the axis of symmetry x=h, and therefore are located at h-d and h+d. (A-REI.4, A-CED.4)
• Secondary Math Three, Module 5, Task 3 “Taking Another Spin”: Students approximate the volume of solids of revolutions whose cross section include curved edges, by replacing them with line segments. Teachers are encouraged to share students' multiple strategies, starting with simple decomposition and ending with a sophisticated one, such as slicing the solid into a stack of circular disks, every ½ unit along the horizontal axis. (G.MG.1, G.GMD.4)
• Secondary Math Three, Module 6, Task 5 “Moving Shadows”: Students continue to use the ideas, strategies, and representations discovered when completing the Ferris wheel tasks from the previous lessons. Students are asked to describe the periodic motion of the rider’s shadow on the Ferris wheel as the shadow moves back and forth across the ground when the sun is directly overhead. Students apply the cosine function to determine the distance horizontally from the center of the wheel and derive the function horizontal position of the shadow = 25cos(18t). Discussions include defining cos(18t) when 18t is not located within the first quadrant. (F-TF.5, F-TF.2, G-SRT.8)

The materials reviewed for Mathematics Vision Project Integrated series meet the expectations for being coherent and making meaningful connections within each course and throughout the series. Overall, the materials include connections as the tasks are reexamined so that familiar mathematical situations are viewed with a new level of sophistication. The sequence of the materials is designed to spiral concepts throughout the entire series.

• Secondary Math One, Module 1, Task 4, “Scott’s Push-Ups” addresses F-BF.1, F-LE.1, F-LE.2, and F-LE.5. Students analyze the pattern of push-ups Scott will include in his workout. This task elicits tables, graphs, and recursive and explicit formulas that focus on how the constant difference shows up in each of the representations and defines the function as an arithmetic sequence. In Secondary Math Two, Module 1, Task 3, “Scott’s Macho March” addresses standards F-BF.1, F-LE.A, A-CED.1 and 2, and F-IF.4 and 5. Students revisit Scott’s workout, but this time his push-up pattern creates a quadratic model. Again, students have the opportunity to use algebraic, numeric, and graphical representation to represent a story with a visual model. In Secondary Math Three, Module 3, Task 1, “Scott’s Macho March Madness” addresses F-BF.1, F-LE.3, and A-CED.2. The purpose of this task is to develop student understanding of how the degree of a polynomial determines the overall rate of change.
• Secondary Math Three, Module 6, Task 6, “Diggin’ It” addresses standards F-TF.1 and F-TF.2. The purpose of this task is to discover alternative ways of measuring a central angle of a circle: in degrees, as a fraction of a complete rotation, or in radians. Students practice using right triangle trigonometry to find the coordinates of points on a circle and use the relationship between arc length measurements and radian angle measurements all within the context of an archeological dig. This task builds upon what students did in Secondary Math Two, showing the length of an arc intercepted by an angle is proportional to the radius and defining the radian measure of the angle as the constant of proportionality (G-C.5). This is followed by Task 7, “Staking It” that addresses standards F-TF.1 and F-TF.2. This task solidifies students’ previous understanding of radians as the ratio of the length of an intercepted arc to the radius of the circle on which that arc lies and uses radian measurement as a proportionality constant in computations.

The materials demonstrate their coherence by revisiting the same contexts and increasing the level of sophistication of the mathematics students engage in.