High School - Gateway 2

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing intentional opportunities for students to develop procedural skills and fluency. Within the lessons, students are provided with opportunities to develop procedures for solving problems that begin to develop fluency. A practice set that includes 10-15 problems is present for each lesson. These practice sets are often “naked number” problems with no context and provide students the opportunity to practice procedural skills.

Some highlights of strong development of procedural skills and fluency include the following:

• A-APR.1: Mathematics II Unit 1 Lesson 1.2.1 and Mathematics III Lesson 2A.1.2 provide opportunities for students to add, subtract, and multiply polynomials. Mathematics III questions extend students’ procedural fluency from those problems students were exposed to in Mathematics II by using larger exponents and more terms in a polynomial expression.

• A-SSE.2: Mathematics II Unit 3 Lesson 3.1 on pages 63-104 reinforces vocabulary and concepts of the parts of expressions and develops skill with writing expressions in different ways in the practice tasks. Mathematics III Unit 2A (pages 46-91) and Unit 2B (pages 5-79), in the Station Activities Set 1, have students build on previous concepts of simplification to rewrite complicated expressions. These stations develop procedural skills as students are required to work from both representations of expressions.

• F-BF.3: There are opportunities provided throughout the series for students to identify the effect of replacing f(x) by f(x)+k, k f(x), f(kx), or f(x + k). Mathematics I Unit 2 Lesson 2.8.2 provides practice with linear and exponential functions; Mathematics II Unit 4 Lessons 2.4.1, 2.4.2, 2.6.1, and 2.6.2 provide practice with quadratic, square root, cube root, and absolute value graphs; and Mathematics III Lessons 3.3.1 and 4B.2.1 provide practice with trigonometric, quadratic, exponential, logarithmic, and linear functions.

• G-GPE.4: Mathematics II Unit 5 Lessons 5.7.1 and 5.7.2 provide opportunities (examples, problem-based task, and practice exercises) for students to use the slope formula, distance formula, and midpoint formula to classify quadrilaterals. Several cases are considered (not only proving a quadrilateral is a parallelogram), and students also work with multiple problems during the practice and guided practices to develop the procedural steps required to prove geometric theorems using coordinates in Mathematics I Unit 6 Lesson 6.1.2.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Students work with mathematical concepts within a real-world context. Each lesson contains a problem-based task at the end of the lesson. The problem-based task includes material found in each lesson in a contextualized situation. Single-step and multi-step contextual problems are used throughout all series’ materials. Non-routine contextual problems are also present within the materials. The problem-based tasks also require greater levels of problem solving sophistication as the series progresses.

Examples of mathematical concepts found in application are as follows:

• G-SRT.8: In Mathematics I Unit 5 Lesson 5.9.3, students use trigonometric functions to solve angle of elevation and depression problems. Practice problems include word problems where students must sketch a diagram of the situation and then solve the problem. The problem-based task for the lesson requires students to complete two trigonometric functions and then subtract to find the answer, therefore creating a multi-step contextual problem. In Mathematics II Unit 5 Lesson 5.8, students are asked to determine the dimensions of a ramp using right triangle trigonometry.

• G-MG.2: In Mathematics III Unit 4B Lesson 4b.5.2, a non-routine contextual problem is located on page 363 as students relate the density of ice to a graph and table and determine an equation to represent the data.

• A-SSE.3: In Mathematics I Unit 1 Lesson 1.2.1 Problem Based Tasks, scaffolding practice, and student practice stress the application of mainly linear relationships. For example, in Guided Practice 3, students must create linear equations to determine when two cars will meet. In Mathematics II Unit 3, the unit has various application problems where station activities and large group discussions provide for application scenarios.

• F-IF.4: In Mathematics I Unit 2 Lesson 2.4, students are asked to use information about the purchase of a car to construct a graph of the value of the car over time and identify key features of the graph.

• A-CED.4: In Mathematics III Unit 4B Lesson 4b.1, students are given formulas which relate the frequency and length of the strings on stringed instruments and asked to create a combined formula to determine the tension on the string.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for the three aspects of rigor being present independently throughout the program materials, and multiple aspects of rigor being engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each lesson includes a common set of components: Warm-Up, Scaffolded Practice, Guided Practice, a Problem-Based Task, Interactive Applets, and Practice Sets. Conceptual development is predominantly addressed independently in tasks specifically called out as Conceptual Tasks. Procedural skills are developed throughout the materials. Engaging applications, although included in the Warm-Up and regularly in a few problems in each Practice Set, are principally addressed in Problem-Based Tasks.

Examples of where the materials independently engage aspects of rigor include:

• In Mathematics I, Unit 2, Lesson 2.4.3, the Interactive Practice Problem GeoGebra “Average Rate of Change” supports the conceptual development of average rate of change.  Students interact with sliders and entry fields to alter the parameters of a function (linear or quadratic) and the interval of change. Students see changes in the graph, slope triangle, and average rate of change calculations. 

• The Mathematics II Program Overview indicates that “activities incorporate concept and skill development and guided practice, then move on to the application of new skills.” Those applications can often be found at the end of the Problem Sets. In Mathematics II, Unit 6, Lesson 6.6.1, Problem Set A, Problem 10, students apply their knowledge of completing the square to find “the geometric description of the region” of the park for which Marco, a park ranger, is responsible. 

• In Mathematics III, Unit 4 [Unit 3], Lesson 1, the Essential Questions focus on concept knowledge and procedures. For example, the list includes “What is a reference angle and how is it found?” and “What are the special angles and how do you find their trigonometric ratios?”  In keeping with these Essential Questions, the Problem Sets in Lessons 3.1.2–3.1.4 predominantly consist of procedural skill practice: students convert between degrees and radians, find reference angles, and find the coordinates for terminal sides of angles.  

Examples of where the materials engage multiple aspects of rigor simultaneously include:

• In Mathematics I, Unit 4, Lesson 4.1.1, students engage with two-way frequency tables in a way that supports their ability to respond to real-world applications. Every table and every question throughout the lesson, including the applets, pertains to a real-world context. Students practice finding marginal and conditional frequencies throughout the lesson. In one of the final Problem Sets (Problem Set B, Problems 1-6), students create a two-way frequency table that shows buildings preferred by residents of each location, find marginal and conditional frequencies, describe trends, and explain how the information could be used to decide where to build each of the three buildings.

• In Mathematics II, Unit 3, Topic B, Conceptual Task, students engage with two student work samples: Asked to solve a quadratic equation, Avi completed the square while Ben used the quadratic formula. Students begin by investigating the two solutions for errors. They answer questions by explaining the student work, supporting the choice of a solution method, explaining what solutions have in common, identifying commonalities between solutions, and describing how the methods might be related.

• In Mathematics III, Unit 6 [Unit 4B], Lesson 4B.3.1, Problem-Based Task, students engage with multiple representations (i.e., data table, equations, and graph) of a function that models levels of carbon dioxide in the atmosphere over a six-decade period. Specifically, students use graphing technology to plot data from a table and compare the key features of the computer-generated equation with the given sine function. In addition, they combine the functions and use the newly created function to make a prediction. Monitoring and Coaching questions included in the Implementation Guide encourage students to interpret key features of the function in terms of the context and explain the disparities in the two graphs. Throughout this task, students apply what they know about sine functions to this real-world scenario.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). Overall, the instructional materials develop both MP1 and MP6 to the full extent of the mathematical practice standards. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. In each of the units there is also a set of station activities that includes a discussion guide. These discussion guides prompt the instructor to ask discussion questions to help students to make sense of the task and to provide responses including precise vocabulary. Emphasis is placed on using units of measure and labeling axes throughout the series (explicit instruction in Mathematics I Lesson 1.2.1 and expectations of using correct units carried throughout the rest of the series). Making sense of answers within the context of a problem is also emphasized. Students also persevere in problem solving in each problem-based task at the end of each lesson.

• Mathematics II Unit 3 Lesson 2.3. (A-SSE.2): Students solve a quadratic equation and determine whether both solutions make sense in the context of a throwing a basketball.

• Mathematics I Unit 2 Lesson 10.1 (F-LE.5): Students need to interpret what the parameters represent in the context of a problem in order to determine whether a solution makes sense. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "check to make sure that students understand how the pricing on the cell phone plans works. Ask them to determine how much someone would pay under each plan for a given number of minutes."

• Mathematics I Unit 5 Lesson 6.1 (G-CO.7): Students identify corresponding parts of congruent triangles as they are introduced to symbolic notation and markings used to represent congruent side and angles in geometric figures. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "encourage students to discuss their prior knowledge of angle pairs formed by a transversal that intersects parallel lines."

• Mathematics III Unit 2 Lesson 2.2: Students use A-REI.2 and the Pythagorean Theorem (G-SRT.8) to solve problems involving radicals. Students are asked to solve applications in both the warmup and problem-based task that require sense-making and perseverance to initiate and precision of units and language to solve.

• Mathematics III Unit 4A Lesson 4a.3: Students work on F-IF.7e. The problem-based task involves creating functions that model the voltage in a three phases of AC coming from a generator. Students will need to persevere to begin the problem and will need to be precise in mathematical language to finish the problem.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards. The Program Overview contains a Correspondence to Standards for Mathematical Practice that focuses on the relevant attributes of Problem-Based Tasks (PBTs). Each PBT “uses a meaningful real-world context that requires students to reason both abstractly about the situation/relationships and quantitatively about the values representing the elements and relationships” (MP2). In addition, because each PBT “provides opportunities for multiple problem-solving approaches and varied solutions, students are required to construct viable arguments to support their approach and answer,” which, in turn, “provides other students the opportunity to analyze and critique their classmates’ reasoning” (MP3). Although the Program Overview focuses on the PBTs, instances of where and how the materials attend to the intentional development of MPs 2 and 3 is not limited to the PBTs.

The materials develop MP2 as students are provided opportunities to develop their mathematical reasoning skills in connection to course-level content across the series. Examples where students reason abstractly and quantitatively include, but are not limited to:

• In Mathematics I, Unit 1, Station Activities, Set 2, Station 4, students match inequalities to real-world situations. After completing the matching task, students explain the strategies used to match the inequalities to the situations. In this activity, students decontextualize a situation to represent it symbolically and contextualize the symbolic representations by considering if the calculated quantities make sense in the given real-world scenarios.

• In Mathematics II, Unit 5, Lesson 5.6.4, the Problem-Based Task Implementation Guide indicates that “students will reason abstractly as they make sense of the information represented in the scenario … and will reason quantitatively as they calculate the midpoints and slopes of each side length of the triangle.”

• In Mathematics III, Unit 5 [Unit 4A], Lesson 4A.2.3, Problem-Based Task, students reason abstractly as they recognize the need to create a natural logarithmic function to describe the exponential growth of global consumer Internet traffic from 2006 to 2014. Students also reason quantitatively as they “substitute the given values from the calculator into the natural logarithmic equation and use the equation to evaluate the number of petabytes per month in a given year.”

The materials attend to the intentional development of MP3 in connection to course-level content across the series. Examples where students construct viable arguments and critique the reasoning of others include, but are not limited to:

• In Mathematics I, Unit 3, Lesson 3.1.1, the Problem-Based Task Implementation Guide indicates: “The focus of the task is to construct a viable argument as to why the answer to the magic number game is always 3. Students will construct their arguments based on their creation of an equation, which is based on following steps and performing calculations on a numerical expression. They will use properties of equality and properties of operations to justify their steps and explanations. Ask students to put their ideas into writing, and encourage students who disagree with each other on any of the steps in the process to discuss and explain their thinking.”

• In Mathematics II, Unit 4, Topic D (online) Learning/Performance Task: Mathematics Assessment Resource Service “Representing Conditional Probabilities 1,” students work together in small groups of two to three to share ideas about the task and plan a joint solution. Materials indicate that the teacher prompts students to listen carefully to explanations and “ask questions if you don’t understand or agree with the method.” Later in the task, students compare different solution methods. The materials indicate that students “compare two arguments and determine correct or flawed logic and prompts students to evaluate peer arguments.”

• In Mathematics III, Unit 6 [Unit 4B], Station Activities: Choosing a Model, students engage with four different activities that support their ability to distinguish between linear, quadratic, and exponential functions. As part of the debrief, students construct viable arguments when they explain how to “distinguish a linear function, an exponential function, and a quadratic function from one another using a table of data.” Implementing the Think-Pair-Share routine provides students with the opportunity to critique the reasoning of others.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

Throughout the materials, the most notable aspect of MP4 is the intentional use of representations to model and to interpret the results of mathematical situations. Examples of where and how students model with mathematics include:

• In Mathematics I, Unit 2, Lesson 2.1.1, Problem-Based Task: Saving for College, students use representations (i.e., equations and graphs) to compare two different methods of compensation: commission-based wages or hourly wages. Given the equation that represents commission-based wages, students formulate an equation to represent hourly wages, graph the two methods of compensation, and describe the earning potential based on the two types of wages.

• In Mathematics II, Unit 2, Lesson 2.1.1, Problem-Based Task: How High Can a Frog Jump?, students represent a mathematical situation with a graph and interpret the results. Specifically, given a quadratic function that models a frog’s height above the water as it jumps across a creek, students determine if it is possible for the frog—with and without jumping—to catch a fly that is “cruising at a height of 5 feet above the water.” The Problem-Based Task Coaching indicates (in order) that students consider and justify if the frog can catch the fly without jumping, indicate and justify whether the extremum is a minimum or maximum, state the vertex, consider if the frog can catch the fly by jumping, and finally sketch the graph of the paths of the frog and the fly. The Implementation Guide indicates that students might opt to sketch the graph of both functions first, then answer the rest of the questions.

• In Mathematics III, Unit 6 [Unit 4B], Lesson 4B.1.1, Guided Practice, students represent mathematics with an equation (in one variable) and interpret the results. Students begin by writing an equation in words to model the total cost of producing personalized cases before creating a variable equation to model the cost of n cases. Using this equation, students then determine how much money will be left to spend on cases after paying a fee and how many cases can be purchased within the context of the scenario. Students use the equation to check the result.

Examples of where and how students use appropriate tools strategically (MP5) include:

• In Mathematics I, Unit 2, Lesson 7, Conceptual Task, students consider questions related to meeting the 1979 World Record for consecutive jumping jacks. Given a pattern of completed jumping jacks in a bar graph, students determine how many jumping jacks will be completed if the pattern of jumps is extended to two weeks. Exploration Questions prompt students to make a table, create a graph, and create an equation to extend the pattern before choosing a method to calculate the number of days it would take to exceed the 1979 World Record of 27,000 consecutive jumping jacks, explaining their method, and extrapolating if the same method would be reasonable beyond a year’s time.

• In Mathematics II, Unit 1, Lesson 1.1.1, Problem-Based Task, students use a given algebraic model to approximate a town’s population eight years from today. Given the complexity of the model, students use a calculator to perform the calculation. Students will be required to use the calculator features strategically to enter the model correctly with rational exponents.

• In Mathematics III, Unit 1, Lesson 1.2.2, Problem-Based Task, students determine if it is reasonable to assume that making eight consecutive foul shots for a player who typically makes 80% of her free throws can be attributed to chance variation alone. Students choose their means (e.g., graphing calculator or deck of cards) of running 20 simulations of a player shooting eight foul shots with an 80% chance of success before calculating the percent of simulations in which all eight shots are made and interpreting the results.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

Math Practice Standards MP7 and MP8 are used to enrich the mathematical content as 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. 

Examples of where and how students look for and make use of structure (MP7) include:

• In Mathematics I, Unit 1, Lesson 1.4.1, Problem-Based Task, students look for and make use of the structure in the information provided about two types of skates to create two linear inequalities in two variables. Students use the system of linear inequalities and constraints of the situation to determine some possible combinations of the number of figure skates and hockey skates that can be made.

• In Mathematics II, Unit 5, Lesson 5.3.1, Problem-Based Task, students use the structure of similar figures to determine the two possible locations for a vertex of a triangle on the coordinate plane.

• In Mathematics III, Unit 3 [Unit 2B], Lesson 2B.1.1, Problem-Based Task, students use structure in the expressions found in the numerator (difference of two squares) and denominator (quadratic trinomial that has linear factors) of a rational expression to write a simplified equivalent expression.

Examples of where and how students look for and express regularity in repeated reasoning (MP8) include:

• In Mathematics I, Unit 5, Lesson 6, Conceptual Task, students determine if all triangles that share a right angle and a common side length (3 in.) are congruent. Students use geometric models and the Pythagorean theorem repeatedly to justify their conclusion. In addition, students recall congruence rules learned earlier in the unit to state conditions under which all triangles are congruent. 

• In Mathematics II, Unit 1, Lesson 1.2.1, Problem-Based Task, students write an expression to represent the perimeter of a small cabin that has both known and unknown dimensions. Students express regularity in repeated reasoning as they simplify the expression for perimeter in terms of x.

• In Mathematics III, Unit 3 [Unit 2B], Lesson 2B.1.3, Problem-Based Task, students calculate the area of a right triangle given rational expressions \frac{3x}{x+1} and \frac{x-1}{x^2} for the lengths of the legs. Students build on their previous knowledge of calculating the area of a triangle with only numerical values, by repeating the same reasoning and process to calculate the area with algebraic expressions.