High School - Gateway 2

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectations of intentionally developing students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Most of the lessons across the series are exploratory in nature and encourage students to develop understanding through questioning and through various activities. Concepts build over many lessons within and between courses in the series. Examples highlighting specific clusters include:

• A-REI.A and B: Secondary Math One, Module 4 builds students’ conceptual knowledge by first introducing multivariable linear equations and then having students express given relationships in equivalent forms. Students engage with inequalities as they encounter the contextual need for inequalities. Students consider the differences and similarities between solving inequalities and solving equations, including that inequalities produce a range of solutions, the inequality symbol must be changed when multiplying or dividing by a negative number, and the reflexive property is true only for equations.
• N-RN.A: In Secondary Math Two, Module 3 a contextual situation offers students the opportunity to understand how values of a dependent variable can exist on the intervals between the whole number values of the independent variable for a continuously increasing exponential function. Next, students examine the role of positive and negative integer exponents and begin to understand the need for rational exponents. Students further develop their conceptual understanding by verifying that the properties of integer exponents remain true for rational exponents.
• A-APR.B: In Secondary Mathematics Three, Module 3 students develop an understanding of multiplicity and a deeper understanding of the relationship between the degree and the number of roots of a polynomial. Then, students use their background knowledge of quadratic functions and end behavior to extend their understanding to higher-order polynomials. The polynomials in this module are factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, the multiplicity of a given root, and what the multiplicity would look like graphically. Finally, students extend their understanding of the Fundamental Theorem of Algebra and the nature of roots by applying the Remainder Theorem.
• G-GPE.5: In Secondary Math One, Module 8, Task 2 students prove that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals. The proofs use the ideas of slope triangles, rotations, and translations and are preceded by a specific case that demonstrates the idea before students are asked to follow the logic using variables.
• G-GPE.6: In Secondary Math Two, Module 6, Task 6 students use similar triangles and proportionality to find the point on a line segment that partitions the segment in a given ratio. Students are first asked to find the midpoint of a segment using two possible strategies and use similar triangles to find segments in ratios other than 1:1. The formula for finding the midpoint of a segment is formalized during the discussion. The discussion can also be extended to derive a formula for finding the midpoint that partitions a segment in any given ratio.
• G-GPE.1, 2: In Secondary Math Two, Module 8, Task 1 students cut out triangles and pin them to a coordinate plane to build a unit circle, effectively developing their understanding of the relationship between the Pythagorean Theorem and the equation of a circle at the origin. Students connect their geometric understanding of circles as the set of all points equidistant from a center to the equation of a circle. This task focuses on a circle (constructed of right triangles) with a radius of 6 inches in order to focus on the Pythagorean theorem and use it to generate the equation of a circle centered at the origin. After constructing a circle at the origin, students consider how the equation would change if the center of the circle is translated.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation to support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications. The materials use real-world situations in which students can apply mathematical concepts, and in situations where a real-world context is not immediately appropriate, the materials begin with abstract situations (graphs, dot models, etc.) and build to the application of the concept in a real-world situation in a later task. Every lesson involves a task, and every task is a real-world situation or a mathematical model that will build to a real-world situation.

The series includes numerous applications across the series, and examples of select standard(s) that specifically relate to applications include, but are not limited to:

• A-CED.3: In Secondary Math One, Module 5, Task 1 the pet-sitting problem uses systems of equations and inequalities to build a business model, minimize costs, and maximize profit.
• F-IF.4,5: In Secondary Math One, Module 3, Task 2 students use tables and graphs to interpret key features of functions (domain and range, where function is increasing/decreasing, x- and y- intercepts, rates of change, discrete vs. continuous) while analyzing the characteristics of a float moving down a river. Students interpret water depth, river speed, and distance traveled using the function skills they are developing.
• F-BF.1: In Secondary Math Two, Module 1, Task 2 students develop a mathematical model for the number of squares in the logo for size n. Students are encouraged to use as many representations as possible for their mathematical model.
• F-TF.5: In Secondary Math Three, Module 6.2 students use the Ferris wheel to determine how high someone will be after 2 seconds, after observing that the Ferris wheel makes one complete rotation counterclockwise every 20 seconds. Students are continuing the work from a previous task in 6.1. Students then determine elapsed time since passing a specific position. Students generate a general formula for finding the height of a rider during a specific time interval and are then asked how they might find the height of the rider for other time intervals.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. Overall, the three aspects are balanced with respect to the standards being addressed. The instructional materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations by giving appropriate attention to developing students’ conceptual understanding, developing procedural skill and fluency, and providing engaging applications.

The materials engage students in each of the aspects of rigor in a pattern that repeats itself throughout that materials. Each module contains Developing Understanding (conceptual understanding), Solidifying Understanding, Practicing Understanding, and the Ready Set Go (procedural skill) activities.

For example, in Secondary Math One, Module 2, Task 1, a Developing Understanding Task focuses on conceptual understanding as students build upon their experiences with exponential and geometric sequences and extend to the broader class of linear and exponential functions with continuous domains. Students compare this variety of functions using various representations (table, graph, and equation). In Task 2, a Solidifying Understanding Task, students discern when it is appropriate to represent a situation with a discrete or continuous model, thus deepening conceptual understanding. This task also has students practice modeling with mathematics by connecting the type of change (linear or exponential) with the nature of that change (discrete or continuous) which develops students’ procedural skill and fluency. Throughout both tasks, problems are presented to students within real-world contexts (medicine metabolized within a dog’s bloodstream, library reshelving efficiency, e-book download rate, savings accounts, pool filling, pool draining, etc), so students are learning the mathematical concepts and procedures while understanding the application of the mathematics.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation that materials support the intentional development of making sense of problems and persevering in solving them, as well as attending to precision (MP1 and MP6) in connection to the high school content standards. Overall, MP1 and MP6 are used to enrich the mathematical content. Throughout the materials, students are expected to make sense of problems and persevere in solving them while attending to precision. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Some examples of MP1 are as follows:

• In Secondary Math One, Module 3, Task 1 students are able to make sense of creating graphs, given a situation. Students are already familiar with graphing rate of change and continuous and non-continuous situations. This task addresses domain and step functions. Students persevere in creating graphs by analyzing what is happening during each interval of time on their graph.
• In Secondary Math Two, Module 3, Task 10 students practice with the arithmetic of irrational and complex numbers and make conjectures as to which of the sets of integers, rational numbers, irrational numbers, real numbers, or complex numbers are closed under the operations of addition, subtraction, and multiplication. Students also experiment with the closure of the set of polynomial functions under the operations of addition, subtraction, and multiplication. Students work with 15 incomplete conjectures and are asked to: “1. Choose the best word to complete each conjecture. 2. After you have made a conjecture, create at least four examples to show why your conjecture is true. 3. If you find a counter-example, change your conjecture to fit your work.” Students creating their own examples and correcting themselves when they find they had initially made the wrong choice provides them the opportunity to persevere.
• In Secondary Math Three, Module 7, Task 2 students make conjectures about how the features of individual functions will show up in the graph of the combined functions when asked to sketch a graph of the path of a rider on a proposed thrill ride at a local theme park. Students then compare their predictions to the actual graphs. Students change the viewing window on their graphing calculator to obtain the information they need to reveal as many of the features of the graphs as possible.

Some examples of MP6 are as follows:

• In Secondary Math One, Module 5, Task 4 students represent constraints with equations or inequalities and with systems of equations and/or inequalities. Students must interpret the solutions as viable or not depending on the context. Students must attend to the language in the constraints. Students convert between units of time and use fractional coefficients within the inequalities, thus also attending to precision of numbers.
• In Secondary Math Two, Module 7, Task 1 students use correct mathematical vocabulary when describing and illustrating their process for finding the center of rotation of a figure consisting of several image/pre-image pairs of points.
• In Secondary Math Three, Module 5, Task 4 students decompose a geometric solid of revolution into familiar three-dimensional objects whose volumes can be calculated. Students calculate the weight of 16d nails, use density information of steel, and complete conversions from ounces to pounds. Students must be precise with their conversions to perform their calculations.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation that materials support the intentional development of reasoning and explaining (MP2 and MP3) in connection to the high school content standards, as required by the MPs. Overall, MP2 and MP3 are used to enrich the mathematical content found in the materials, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Some examples of MP2 are as follows:

• In Secondary Math One, Module 5, Task 7, “Get to the Point," students reason abstractly and quantitatively which window cleaning company to hire, by using a table, a graph, and/or algebra.
• In Secondary Math Two, Module 2, Task 1, “Transformers: Shifty y’s,” students reason abstractly and quantitatively, relating the numeric results in the tables to the graphs to explain why the graphs are transformed as they are.
• In Secondary Math Three, Module 3, Task 2, “Which is Greater?” students reason abstractly and quantitatively about the rates of change and end behavior when comparing various one-variable expressions, by examining how the order changes, arranging either least to greatest or greatest to least depending on the values of x close to negative infinity, zero, and positive infinity. Students reason quantitatively by substituting in values and reasoning abstractly, by making assumptions based on their knowledge of exponents, comparing polynomial to exponential functions, and comparing what happens when the degree of the polynomial is even or odd when values of x approach −∞.

Some examples of MP3 are as follows:

• In Secondary Math One, Module 3, Task 2, “Floating Down the River,” students explain why they either agree or disagree with each observation Sierra made. Some of Sierra’s observations include: “a) The depth of the water increases and decreases throughout the 120 minutes of floating down the river, b) The distance traveled is always increasing, or c) The distance traveled is a function of time.”
• In Secondary Math Two, Module 6, Task 3, “Similar Triangles and Other Figures,” students read through Mia and Mason’s conjectures about similar polygons and decide which they believe are true. Students are also presented “explanations” from either Mia or Mason and must write an argument deciding whether they agree.
• In Secondary Math Three, Module 4, Task 5, “Watching your Behavior,” students work with partners to try to come up with various rational functions that create different end behaviors. For each rational function they create, they state the end behavior and come up with an equation that models the end behavior asymptotes. With examples, they provide evidence that their end behavior is correct and begin to identify generalizations to find the end behavior asymptotes for various rational functions.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation that materials support the intentional development of addressing mathematical modeling and using tools (MP4 and MP5), in connection to the high school content standards, as required by the MP. Overall, MP4 and MP5 are used to enrich the mathematical content, and these practices are not treated as isolated experiences for the students. Throughout the materials, students model with mathematics and use tools strategically. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Some examples of MP4 are as follows:

• In Secondary Math One, Module 5, Task 3 students use systems of equations, tables and graphs to model the start-up costs of a new business and use the information to minimize costs and maximize profit.
• In Secondary Math Two, Module 9, Task 3 students use Venn diagrams to model the situation, analyze the data, write various probability statements (unions, intersections, and complements), and then apply the Addition Rule and interpret the answer in terms of the model.
• In Secondary Math Three, Module 5, Task 3 students examine a solid of revolution and a frustum to create a strategy for finding volume. Students use a variety of strategies to decompose a figure that consists of curved edges into cylinders, frustums, and cones in order to generate a sequence of better approximations of the actual volume of the solid.

Some examples of MP5 are as follows:

• In Secondary Math One, Module 9, Task 5 students use technology such as the graphing calculator, GeoGebra, or Desmos to compute and interpret the correlation coefficient of a linear fit.
• In Secondary Math Two, Module 2, Task 1 students use technology to explore the results of various changes to the function they are investigating. Students choose the technology.
• In Secondary Math Three, Module 2, Task 5 students choose to use calculators or other technology with base 10 logarithmic and exponential functions to complete the problems.

The instructional materials reviewed for Mathematics Vision Project Integrated series meet the expectation that materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the MP. Overall, MP7 and MP8 are used to enrich the mathematical content, and these practices are not treated as isolated experiences for the students. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

The materials frequently take a task from a previous course and add a new contextual layer to the mathematics, such as the Pet Sitter Task and Bruno Bites Task. Students are constantly extending the structures used when solving problems that build on one another and, as a result, are able to solve increasingly complex problems. In the instructional materials, repeated reasoning based on similar structures allows for increasingly complex mathematical concepts to be developed from simpler ones.

Some examples of MP7 are as follows:

• In Secondary Math One, Module 2, Task 1 builds upon students’ previous experiences with arithmetic and geometric sequences to extend to the broader class of linear and exponential functions with continuous domains. Students use tables, graphs, and equations to create mathematical models for contextual situations. Students continue to define linear and exponential functions by their patterns of growth. Students are repeatedly asked to identify similarities and differences between problems in an effort for them to identify the structure present.
• In Secondary Math Two, Module 4, Task 3 students learn how to graph, write, and create linear absolute value functions by looking at structure and making sense of piecewise-defined functions. They connect prior understandings of transformations, domain, linear functions, and piecewise functions and share strategies for how to go from one representation to another to graph and write equations for absolute value piecewise functions.
• In Secondary Mathematics Three, Module 3, Task 6, “Seeing Structure,” students use their background knowledge of quadratic functions and end behavior to extend their understanding of polynomials in general. The polynomials in this task are easily factorable and allow students opportunities to solidify their understanding of end behavior, the Fundamental Theorem of Algebra, and the multiplicity of a given root (and what that would look like graphically).

Some examples of MP8 are as follows:

• In Secondary Math One, Module 5, Task 6, students practice solving systems of linear inequalities by identifying the overlapping region of the half-planes that form the solution sets of each of the two-variable inequalities in the system. Students recognize the difference between a strict inequality and one that includes the points on the boundary line as part of the solution set. Through repeated practice students develop a procedure for solving a system of linear inequalities.
• In Secondary Math Two, Module 3, Task 5, students use what they already know about quadratic functions to generalize a process for finding x-intercepts for any quadratic function that has them. Students use the method of completing the square to rearrange the formula to highlight a quantity of interest, using the same reasoning as in solving equations.
• In Secondary Math Three, Module 6, Task 7, students calculate the x- and y- coordinates for stakes placed on a circle, as well as the arc length on concentric circles placed around an archeological site. Repeating the same calculations, students recognize they can just double or triple the coordinates or arc length given on the 10-meter circle to get the coordinates or arc length on the 20-meter or 30-meter circles.